{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0001","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2eddaef371d53aefb1b3076cdd950c1b63d99f26aed7cfe5206a906b41da28","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214001,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-10,-3,-2],\"x0\":-2,\"y0\":-11}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-10, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-11. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-10, -3, -2], -2, -12, -11, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -3, -2], -2, -12, -11, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0002","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aba693b139a3912e13668749d7783a181ee2764a21e5765442d15e155314c812","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214002,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-9,-2,0],\"x0\":-1,\"y0\":-7}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-9, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-9, -2, 0], -1, -7, -7, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, -2, 0], -1, -7, -7, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0003","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43013c2d4b34d4578b2f91d71bcbb456718ccd008ea90ad54cc85f4814c309b6","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214003,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-8,-1,2],\"x0\":0,\"y0\":-8}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-8, -1, 2], 0, -8, -8, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, -1, 2], 0, -8, -8, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0004","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4328b85164f6292b644f19433ccb909070027e9f6932690853e760306c5a55f","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214004,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-7,0,-1],\"x0\":1,\"y0\":-8}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-7, 0, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-7, 0, -1], 1, -8, -8, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 0, -1], 1, -8, -8, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0005","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"170bc2f87ca2e9b924562cab3e44c7f92f621db34cf88903d1471eb4c94be4bf","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214005,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-6,1,1],\"x0\":2,\"y0\":1}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-6, 1, 1], 2, 0, 1, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 1, 1], 2, 0, 1, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0006","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28775c781684373ecc93ee1e2b2ad2ce7164cdbda88cec7af4e74ea96dfb000a","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214006,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-5,2,-2],\"x0\":-2,\"y0\":-17}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-5, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-17. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-5, 2, -2], -2, -17, -17, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, 2, -2], -2, -17, -17, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0007","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac8fff3d9f6a5989858e56a410130b0549df0230c2617dd85446c891a8c14572","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214007,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-4,3,0],\"x0\":-1,\"y0\":-7}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-4, 3, 0], -1, -7, -7, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, 3, 0], -1, -7, -7, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0008","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"901f8288edf6d743f4245f9311abebe8e8c73e71886a4b77b99370a848958105","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214008,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-3,-3,2],\"x0\":0,\"y0\":-3}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-3, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-3. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-3, -3, 2], 0, -3, -3, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -3, 2], 0, -3, -3, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0009","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a4bc0272963201b7ce8c964afab62847b7e93756f4c703b2f711b3aba35ad94","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214009,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-2,-2,-1],\"x0\":1,\"y0\":-4}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-2, -2, -1], 1, -5, -4, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, -2, -1], 1, -5, -4, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0010","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4480a4fd228f483db9fdd5d1b543f06bbf58eb39b34d27c8fa9c53ded9c984","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214010,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[-1,-1,1],\"x0\":2,\"y0\":1}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([-1, -1, 1], 2, 1, 1, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, -1, 1], 2, 1, 1, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0011","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b803fdee005e0f616aa966b30d5d360dfc37af518371279bcf1b2f6bd6da3e","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214011,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[0,0,-2],\"x0\":-2,\"y0\":-8}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[0, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([0, 0, -2], -2, -8, -8, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0, -2], -2, -8, -8, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0012","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da6c07288bbc89b44852324abded998dcfbb277fadb8f7b661ff242f15973bd","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214012,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[1,1,0],\"x0\":-1,\"y0\":0}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([1, 1, 0], -1, 0, 0, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1, 0], -1, 0, 0, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0013","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe34a5a352d73cf80688f87e2851b626538a0d066cc38baed56cfbf9e28e716","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214013,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[2,2,2],\"x0\":0,\"y0\":3}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[2, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=3. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([2, 2, 2], 0, 2, 3, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2, 2], 0, 2, 3, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0014","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d167a00b27cee70faf42af1bb81520bc1e2806877b09c8d638cb97389da9c933","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214014,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[3,3,-1],\"x0\":1,\"y0\":5}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[3, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=5. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([3, 3, -1], 1, 5, 5, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3, -1], 1, 5, 5, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0015","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba155a62e90c02a7dd94ae8d958644863a864c234e7edd93942eca4384e867b","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214015,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[4,-3,1],\"x0\":2,\"y0\":2}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[4, -3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=2. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([4, -3, 1], 2, 2, 2, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -3, 1], 2, 2, 2, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0016","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"941786246a6624bd49d74bf334804f2cad17c2280df5ac7235342214abf1b5d1","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214016,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[5,-2,-2],\"x0\":-2,\"y0\":1}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[5, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([5, -2, -2], -2, 1, 1, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, -2, -2], -2, 1, 1, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0017","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7edac601a1ec1062261fd1b02b5127a8da22f0fabf1130f8226587b7afa262e8","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214017,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[6,-1,0],\"x0\":-1,\"y0\":8}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[6, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([6, -1, 0], -1, 7, 8, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, -1, 0], -1, 7, 8, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0018","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2d5b65550eb9584af9154418e50e25252b208dc2c814ac7939520ae6a34cc79","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214018,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[7,0,2],\"x0\":0,\"y0\":7}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=7. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([7, 0, 2], 0, 7, 7, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 0, 2], 0, 7, 7, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0019","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e491cd47cf61dbc835376c5da1beee64b16c06ab9bc432ce976315e14630ca5","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214019,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([8, 1, -1], 1, 8, 8, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 1, -1], 1, 8, 8, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0020","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43dc2380c6aed896fee36323f3e66dd21509af1f299d200810f0cab34b4b41b1","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214020,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[9,2,1],\"x0\":2,\"y0\":17}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[9, 2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=17. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([9, 2, 1], 2, 17, 17, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, 2, 1], 2, 17, 17, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0021","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e2d4c06867599b07d9fa40643eebd02cfa63e7459d15039da99d81e91055509","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214021,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[10,3,-2],\"x0\":-2,\"y0\":-3}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[10, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-3. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([10, 3, -2], -2, -4, -3, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, 3, -2], -2, -4, -3, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0022","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44296035d95214a081c251cd5aa28166bc8349ce22da04ad7bfaaed40d242f0b","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214022,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[11,-3,0],\"x0\":-1,\"y0\":14}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[11, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=14. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([11, -3, 0], -1, 14, 14, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -3, 0], -1, 14, 14, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0023","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878fb8738279dac115a333aef0f122bb9544f423a1d96118dadb09db954a03e9","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214023,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[12,-2,2],\"x0\":0,\"y0\":12}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[12, -2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=12. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([12, -2, 2], 0, 12, 12, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, -2, 2], 0, 12, 12, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0024","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c7f56f8fe7633855ce6568d105f95622131adb03335cebd02d5f4f4aa059c4e","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214024,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[13,-1,-1],\"x0\":1,\"y0\":11}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[13, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=11. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([13, -1, -1], 1, 11, 11, Yes)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([13, -1, -1], 1, 11, 11, Yes)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0025","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ec0cf694a97f5f32dab928beca4c836f3748dc4e56c267850adaf39e27f626c","family_id":"qf-app21-ivp-first-order-initial-audit","seed":202610214025,"semantic_template":"qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[14,0,1],\"x0\":2,\"y0\":19}"},"fields":{"title":"An Initial Value Check Is Not an ODE Check","stem":"Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19. Give (coefficients,x0,y(x0),required y0,match?).","answer":{"kind":"expression","value":"([14, 0, 1], 2, 18, 19, No)","explanation":"Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([14, 0, 1], 2, 18, 19, No)."},"skills":["An Initial Value Check Is Not an ODE Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0026","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"194110a854498daacdd4a445a813422c7273cea6968a6ed66319b37c7113e712","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214026,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-10,-3,-2],\"dy0\":4,\"x0\":-2,\"y0\":-11}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-10, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-11, required y_prime0=4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-10, -3, -2], -2, -12, 5, -11, 4, No, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -3, -2], -2, -12, 5, -11, 4, No, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0027","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acf4acd90571f1f22be9a66c0189f253275721ef95e3e443fc26900f62f65f48","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214027,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-9,-2,0],\"dy0\":-2,\"x0\":-1,\"y0\":-7}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-9, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-9, -2, 0], -1, -7, -2, -7, -2, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, -2, 0], -1, -7, -2, -7, -2, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0028","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9825662274544ec48d59b96b20ea0334c2eff88f1e089a0b9cfb10b97c6bed","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214028,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-8,-1,2],\"dy0\":-1,\"x0\":0,\"y0\":-8}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-8, -1, 2], 0, -8, -1, -8, -1, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, -1, 2], 0, -8, -1, -8, -1, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0029","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a77a5c9bf003043efc11788ffe5802e5615d17cd6a4890baa6c52eb86fe3307f","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214029,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-7,0,-1],\"dy0\":-3,\"x0\":1,\"y0\":-8}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-7, 0, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-8, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-7, 0, -1], 1, -8, -2, -8, -3, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 0, -1], 1, -8, -2, -8, -3, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0030","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f09fdea1e93b19ae5126333758ad78ea43a6e285dd49bcfcc8ed190c969192ae","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214030,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-6,1,1],\"dy0\":5,\"x0\":2,\"y0\":1}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1, required y_prime0=5. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-6, 1, 1], 2, 0, 5, 1, 5, No, Yes, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 1, 1], 2, 0, 5, 1, 5, No, Yes, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0031","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b38287aeba080a46f39849005311980532e4d890aefe182bbf66bfd1e4e62798","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214031,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-5,2,-2],\"dy0\":10,\"x0\":-2,\"y0\":-17}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-5, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-17, required y_prime0=10. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-5, 2, -2], -2, -17, 10, -17, 10, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, 2, -2], -2, -17, 10, -17, 10, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0032","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa65a45797d61f6a9717fb3e253e18b77dfa10155ce44c6421356fe61d5e2f63","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214032,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-4,3,0],\"dy0\":2,\"x0\":-1,\"y0\":-7}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-4, 3, 0], -1, -7, 3, -7, 2, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, 3, 0], -1, -7, 3, -7, 2, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0033","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"298f87ce3290d93d8641617469aa851fb864f963b5de3d65581d9eb0944823c6","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214033,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-3,-3,2],\"dy0\":-3,\"x0\":0,\"y0\":-3}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-3, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-3, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-3, -3, 2], 0, -3, -3, -3, -3, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -3, 2], 0, -3, -3, -3, -3, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0034","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62d47de50314c8ff439bc6cc37f98c43ce2f4951685396f76ffb538143c5a522","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214034,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-2,-2,-1],\"dy0\":-4,\"x0\":1,\"y0\":-4}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-2, -2, -1], 1, -5, -4, -4, -4, No, Yes, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, -2, -1], 1, -5, -4, -4, -4, No, Yes, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0035","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6865f010a316a6470838b200d6034827e8b8955aaa180bb0d48ffd657f4e8ddd","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214035,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[-1,-1,1],\"dy0\":2,\"x0\":2,\"y0\":1}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([-1, -1, 1], 2, 1, 3, 1, 2, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, -1, 1], 2, 1, 3, 1, 2, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0036","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"137c345020fa3316532114a4b9e2a9eff4c1b68b1926244db6afcba2d8b12da2","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214036,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[0,0,-2],\"dy0\":8,\"x0\":-2,\"y0\":-8}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[0, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-8, required y_prime0=8. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([0, 0, -2], -2, -8, 8, -8, 8, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0, -2], -2, -8, 8, -8, 8, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0037","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e274dd81e7e71ae1c4e06a244edaad3655e00b1b2338c4bc9e7a90138282c8b","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214037,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[1,1,0],\"dy0\":1,\"x0\":-1,\"y0\":0}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([1, 1, 0], -1, 0, 1, 0, 1, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1, 0], -1, 0, 1, 0, 1, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0038","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02a0f3a8449491c823198a1507c1d6139946f919fc85efa0633b38df1a8626f","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214038,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[2,2,2],\"dy0\":1,\"x0\":0,\"y0\":3}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[2, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=3, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([2, 2, 2], 0, 2, 2, 3, 1, No, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2, 2], 0, 2, 2, 3, 1, No, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0039","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dc87b4a82e4e8d1409470a4e601e6d0d132369c723fd4acbdb0efbf2d3f3d73","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214039,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[3,3,-1],\"dy0\":1,\"x0\":1,\"y0\":5}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[3, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=5, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([3, 3, -1], 1, 5, 1, 5, 1, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3, -1], 1, 5, 1, 5, 1, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0040","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2aae5c07677abd3a3a20a1bee8d53cafcf3eb6074a65c37c09f9ce5df21878c","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214040,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[4,-3,1],\"dy0\":1,\"x0\":2,\"y0\":2}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[4, -3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=2, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([4, -3, 1], 2, 2, 1, 2, 1, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -3, 1], 2, 2, 1, 2, 1, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0041","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c8d970e095641d61330857e1319b42781449a936a6c4fee7e503c768309bfd8","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214041,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[5,-2,-2],\"dy0\":5,\"x0\":-2,\"y0\":1}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[5, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=1, required y_prime0=5. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([5, -2, -2], -2, 1, 6, 1, 5, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, -2, -2], -2, 1, 6, 1, 5, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0042","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c354554f2746ae92c6d04e7ab64bb8e1a468e037f4efab57ecfddfba44641d8b","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214042,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[6,-1,0],\"dy0\":-1,\"x0\":-1,\"y0\":8}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[6, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([6, -1, 0], -1, 7, -1, 8, -1, No, Yes, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, -1, 0], -1, 7, -1, 8, -1, No, Yes, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0043","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"552796a107550c819e7ba05f3f863f48e86f4de36558503c250afed499fd783f","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214043,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[7,0,2],\"dy0\":0,\"x0\":0,\"y0\":7}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=7, required y_prime0=0. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([7, 0, 2], 0, 7, 0, 7, 0, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 0, 2], 0, 7, 0, 7, 0, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0044","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dffa53dc4d1f56b41712e284c37c51fcf8f5246948e6039b0ce9f7824ac7e53","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214044,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[8,1,-1],\"dy0\":-2,\"x0\":1,\"y0\":8}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([8, 1, -1], 1, 8, -1, 8, -2, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 1, -1], 1, 8, -1, 8, -2, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0045","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f41ad61361eb28a5712568a267890e0dc1f2b988d58be27ac97bddfdece598c5","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214045,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[9,2,1],\"dy0\":6,\"x0\":2,\"y0\":17}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[9, 2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=17, required y_prime0=6. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([9, 2, 1], 2, 17, 6, 17, 6, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, 2, 1], 2, 17, 6, 17, 6, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0046","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9dac6dc73d5c0fae1f58638c50e6d9f4028fcb75475ffc325df099c20c8c1f3","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214046,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[10,3,-2],\"dy0\":11,\"x0\":-2,\"y0\":-3}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[10, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-3, required y_prime0=11. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([10, 3, -2], -2, -4, 11, -3, 11, No, Yes, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, 3, -2], -2, -4, 11, -3, 11, No, Yes, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0047","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13927ad6bb653b23059d6c49181cd7f1f2fef063aadb3081e8f88fc954d5ed57","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214047,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[11,-3,0],\"dy0\":-4,\"x0\":-1,\"y0\":14}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[11, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=14, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([11, -3, 0], -1, 14, -3, 14, -4, Yes, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -3, 0], -1, 14, -3, 14, -4, Yes, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0048","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b36bc7e2ed0333171fa6cf414b788f2ce47760c3987a22b8ec75a0068ba9b1","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214048,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[12,-2,2],\"dy0\":-2,\"x0\":0,\"y0\":12}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[12, -2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=12, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([12, -2, 2], 0, 12, -2, 12, -2, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, -2, 2], 0, 12, -2, 12, -2, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0049","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"667b559be7d27023de93c976832049143c4807c5d817a0f972e5a33e8b2d5908","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214049,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[13,-1,-1],\"dy0\":-3,\"x0\":1,\"y0\":11}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[13, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=11, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([13, -1, -1], 1, 11, -3, 11, -3, Yes, Yes, Yes)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([13, -1, -1], 1, 11, -3, 11, -3, Yes, Yes, Yes)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0050","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830f388764c745226b2e45cc92dab5678fb03a70fdf7e9b8eabfa376d92d85d5","family_id":"qf-app21-ivp-second-order-initial-audit","seed":202610214050,"semantic_template":"qf-app21-ivp-second-order-initial-audit/{\"coefficients\":[14,0,1],\"dy0\":3,\"x0\":2,\"y0\":19}"},"fields":{"title":"Value and Slope Initial Data as Separate Conditions","stem":"Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19, required y_prime0=3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).","answer":{"kind":"expression","value":"([14, 0, 1], 2, 18, 4, 19, 3, No, No, No)","explanation":"Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([14, 0, 1], 2, 18, 4, 19, 3, No, No, No)."},"skills":["Value and Slope Initial Data as Separate Conditions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0051","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31f7a7184748ef188924a531318f3d9fc4b4ea05aa4089b947923e4c2cedf1e6","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214051,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-12,-3],\"p\":0,\"q\":-2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-5}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-12, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-2, required y0=-5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, No], [0, -12, -3, -1, No], [1, -15, -3, -1, No]], -2, -6, -5, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, No], [0, -12, -3, -1, No], [1, -15, -3, -1, No]], -2, -6, -5, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0052","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef9a073b5ce74fdf76d989ef3bc74f0a864beca08b5803370582ae424b7ae195","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214052,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-11,-2],\"p\":0,\"q\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-9}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-11, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-1, required y0=-9. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, Yes], [0, -11, -2, 0, Yes], [1, -13, -2, 0, Yes]], -1, -9, -9, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, Yes], [0, -11, -2, 0, Yes], [1, -13, -2, 0, Yes]], -1, -9, -9, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0053","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38199491487615bb2cec4314c6596084dbda06fd2675a276dc56144a84cb5fe2","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214053,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-10,-1],\"p\":0,\"q\":-1,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-10}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-10, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=0, required y0=-10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, Yes], [0, -10, -1, 0, Yes], [1, -11, -1, 0, Yes]], 0, -10, -10, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, Yes], [0, -10, -1, 0, Yes], [1, -11, -1, 0, Yes]], 0, -10, -10, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0054","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"260d72ae80c29361141ac18f15d54aa19c016b1dd6a898756f1a44c31b7fbb26","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214054,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-9,0],\"p\":0,\"q\":0,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-8}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-9, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=1, required y0=-8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, Yes], [0, -9, 0, 0, Yes], [1, -9, 0, 0, Yes]], 1, -9, -8, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, Yes], [0, -9, 0, 0, Yes], [1, -9, 0, 0, Yes]], 1, -9, -8, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0055","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a7c0c1180b7e79f2f8c789779619157ad0a5b7d166cab2be31891bd83dc0b43","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214055,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-8,1],\"p\":0,\"q\":2,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-6}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-8, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=2, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, No], [0, -8, 1, -1, No], [1, -7, 1, -1, No]], 2, -6, -6, Yes, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, No], [0, -8, 1, -1, No], [1, -7, 1, -1, No]], 2, -6, -6, Yes, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0056","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5178c34e1dc78936d73daf536be10b5ba27dc954102d75efd8b3ca4509311e98","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214056,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-7,2],\"p\":0,\"q\":2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-11}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-7, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=-2, required y0=-11. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, Yes], [0, -7, 2, 0, Yes], [1, -5, 2, 0, Yes]], -2, -11, -11, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, Yes], [0, -7, 2, 0, Yes], [1, -5, 2, 0, Yes]], -2, -11, -11, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0057","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db1289a74cba2f296b684a64fe5c3c5af57792bb52c7ba8a5fc998a56050ac38","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214057,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-6,3],\"p\":0,\"q\":3,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-8}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-6, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=-1, required y0=-8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, Yes], [0, -6, 3, 0, Yes], [1, -3, 3, 0, Yes]], -1, -9, -8, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, Yes], [0, -6, 3, 0, Yes], [1, -3, 3, 0, Yes]], -1, -9, -8, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0058","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e9f8fef977ac8fcbc70df8f79a535b6a51f90fd5650775a47506be41dbb8fd2","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214058,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-5,-3],\"p\":0,\"q\":-3,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-5}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-5, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=0, required y0=-5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, Yes], [0, -5, -3, 0, Yes], [1, -8, -3, 0, Yes]], 0, -5, -5, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, Yes], [0, -5, -3, 0, Yes], [1, -8, -3, 0, Yes]], 0, -5, -5, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0059","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76f0d7045cc450aada05753120b18c87d6dd1929ba95a3c6b7a11f03bd01a4ef","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214059,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-4,-2],\"p\":0,\"q\":-1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-6}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-4, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, No], [0, -4, -2, -1, No], [1, -6, -2, -1, No]], 1, -6, -6, Yes, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, No], [0, -4, -2, -1, No], [1, -6, -2, -1, No]], 1, -6, -6, Yes, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0060","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"636bc649382baab385ace80f36ed80451bf543ccdf8d5b4c1ca9be8e2eceac33","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214060,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-3,-1],\"p\":0,\"q\":-1,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-4}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=2, required y0=-4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, Yes], [0, -3, -1, 0, Yes], [1, -4, -1, 0, Yes]], 2, -5, -4, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, Yes], [0, -3, -1, 0, Yes], [1, -4, -1, 0, Yes]], 2, -5, -4, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0061","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ae756844e69e5a0f56e326b4e6870cca48db669aba0d14a41133b4a1c1a943","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214061,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-2,0],\"p\":0,\"q\":0,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-2}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=-2, required y0=-2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, Yes], [0, -2, 0, 0, Yes], [1, -2, 0, 0, Yes]], -2, -2, -2, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, Yes], [0, -2, 0, 0, Yes], [1, -2, 0, 0, Yes]], -2, -2, -2, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0062","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"350e41c75524b6b76746744e96bdfe94918a01b9685b54a0a459003385d7a3b2","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214062,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[-1,1],\"p\":0,\"q\":1,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-2}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=-1, required y0=-2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, Yes], [0, -1, 1, 0, Yes], [1, 0, 1, 0, Yes]], -1, -2, -2, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, Yes], [0, -1, 1, 0, Yes], [1, 0, 1, 0, Yes]], -1, -2, -2, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0063","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ed39046170c4b5463eaaefc259553d420d07d0012add5faa2fc0d971de13ff8","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214063,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[0,2],\"p\":0,\"q\":3,\"x0\":0,\"xs\":[-1,0,1],\"y0\":1}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=0, required y0=1. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, No], [0, 0, 2, -1, No], [1, 2, 2, -1, No]], 0, 0, 1, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, No], [0, 0, 2, -1, No], [1, 2, 2, -1, No]], 0, 0, 1, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0064","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63061831a9a4a16ffcca044822bfa01bade7b4473d33528393c2fdd922af0960","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214064,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[1,3],\"p\":0,\"q\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":4}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[1, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=1, required y0=4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, Yes], [0, 1, 3, 0, Yes], [1, 4, 3, 0, Yes]], 1, 4, 4, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, Yes], [0, 1, 3, 0, Yes], [1, 4, 3, 0, Yes]], 1, 4, 4, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0065","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1499f190839a12aed2106dfd3061fe12531ecda85ee51745d9414a2658f533b4","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214065,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[2,-3],\"p\":0,\"q\":-3,\"x0\":2,\"xs\":[-1,0,1],\"y0\":-4}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[2, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=2, required y0=-4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, Yes], [0, 2, -3, 0, Yes], [1, -1, -3, 0, Yes]], 2, -4, -4, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, Yes], [0, 2, -3, 0, Yes], [1, -1, -3, 0, Yes]], 2, -4, -4, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0066","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dab30397b2b965434c3ee4ec9ad692106634c4d52fe525ae257850f36b9758b","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214066,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[3,-2],\"p\":0,\"q\":-2,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":8}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-2, required y0=8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, Yes], [0, 3, -2, 0, Yes], [1, 1, -2, 0, Yes]], -2, 7, 8, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, Yes], [0, 3, -2, 0, Yes], [1, 1, -2, 0, Yes]], -2, 7, 8, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0067","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"819082b83dabb7b660f3e70d631298a2011fc6cc0e64bd6124de8b883958a8f4","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214067,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[4,-1],\"p\":0,\"q\":0,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":5}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[4, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=-1, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, No], [0, 4, -1, -1, No], [1, 3, -1, -1, No]], -1, 5, 5, Yes, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, No], [0, 4, -1, -1, No], [1, 3, -1, -1, No]], -1, 5, 5, Yes, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0068","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"328b5b91d9bb288b5ebda4e4364beada9d52a42786f53fe6dc211b079e1e7224","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214068,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[5,0],\"p\":0,\"q\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":5}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[5, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=0, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, Yes], [0, 5, 0, 0, Yes], [1, 5, 0, 0, Yes]], 0, 5, 5, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, Yes], [0, 5, 0, 0, Yes], [1, 5, 0, 0, Yes]], 0, 5, 5, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0069","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f692c8d31aca5353671d6362571f5162524cec311c3259be29342562523efdc5","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214069,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[6,1],\"p\":0,\"q\":1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":8}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[6, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=1, required y0=8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, Yes], [0, 6, 1, 0, Yes], [1, 7, 1, 0, Yes]], 1, 7, 8, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, Yes], [0, 6, 1, 0, Yes], [1, 7, 1, 0, Yes]], 1, 7, 8, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0070","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a05ea19a41a66a831bfa3050632059d9f0b31d3338150df96067eb86093ac5e","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214070,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[7,2],\"p\":0,\"q\":2,\"x0\":2,\"xs\":[-1,0,1],\"y0\":11}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[7, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=2, required y0=11. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, Yes], [0, 7, 2, 0, Yes], [1, 9, 2, 0, Yes]], 2, 11, 11, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, Yes], [0, 7, 2, 0, Yes], [1, 9, 2, 0, Yes]], 2, 11, 11, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0071","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5bbecb9eb310faecf0b175bdf0fa883154e4e9ae7dc5d713f947b019f254fe6","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214071,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[8,3],\"p\":0,\"q\":4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":2}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[8, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=4; input grid=[-1, 0, 1], x0=-2, required y0=2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, No], [0, 8, 3, -1, No], [1, 11, 3, -1, No]], -2, 2, 2, Yes, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, No], [0, 8, 3, -1, No], [1, 11, 3, -1, No]], -2, 2, 2, Yes, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0072","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff44c0e39de8245792be3a6541013cde57e7112f941bf70b067eced37405f9da","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214072,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[9,-3],\"p\":0,\"q\":-3,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":13}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[9, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=-1, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, Yes], [0, 9, -3, 0, Yes], [1, 6, -3, 0, Yes]], -1, 12, 13, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, Yes], [0, 9, -3, 0, Yes], [1, 6, -3, 0, Yes]], -1, 12, 13, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0073","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93b7b4e5c7317a1c3934c69421e951335b4a97d836680e27485d342924078334","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214073,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[10,-2],\"p\":0,\"q\":-2,\"x0\":0,\"xs\":[-1,0,1],\"y0\":10}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[10, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=0, required y0=10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, Yes], [0, 10, -2, 0, Yes], [1, 8, -2, 0, Yes]], 0, 10, 10, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, Yes], [0, 10, -2, 0, Yes], [1, 8, -2, 0, Yes]], 0, 10, 10, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0074","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f644c5ecfefe01f2202e76f9b8434718bddd80459be82d586daf3b846d739865","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214074,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[11,-1],\"p\":0,\"q\":-1,\"x0\":1,\"xs\":[-1,0,1],\"y0\":10}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[11, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, Yes], [0, 11, -1, 0, Yes], [1, 10, -1, 0, Yes]], 1, 10, 10, Yes, Yes)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, Yes], [0, 11, -1, 0, Yes], [1, 10, -1, 0, Yes]], 1, 10, 10, Yes, Yes)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0075","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22e102b11533e98abff33d4e14721e1ddfa092b2c0a6075f79f2228eac30c440","family_id":"qf-app21-ivp-first-order-candidate-grid","seed":202610214075,"semantic_template":"qf-app21-ivp-first-order-candidate-grid/{\"coefficients\":[12,0],\"p\":0,\"q\":1,\"x0\":2,\"xs\":[-1,0,1],\"y0\":13}"},"fields":{"title":"A First-Order Candidate: Grid Residuals and Initial Value","stem":"Use the integer polynomial candidate with ascending-power coefficients=[12, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=2, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).","answer":{"kind":"expression","value":"([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, No], [0, 12, 0, -1, No], [1, 12, 0, -1, No]], 2, 12, 13, No, No)","explanation":"Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, No], [0, 12, 0, -1, No], [1, 12, 0, -1, No]], 2, 12, 13, No, No)."},"skills":["A First-Order Candidate: Grid Residuals and Initial Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0076","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9eacec231c1cdb5e3576ac0fbebbf32fd10bc9829f13191632669dfa10a3651","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214076,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-12,-3,-2],\"dy0\":4,\"rhs\":-3,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-13}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-12, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-3; input grid=[-1, 0, 1], x0=-2, required y0=-13, required y_prime0=4. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, No], [0, -12, -3, -4, -1, No], [1, -17, -7, -4, -1, No]], -2, -14, 5, -13, 4, No, No, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, No], [0, -12, -3, -4, -1, No], [1, -17, -7, -4, -1, No]], -2, -14, 5, -13, 4, No, No, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0077","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0257c4995496adaca9a2a7ea2d42c6cb5286f253fe273e743741e3aa7e323a68","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214077,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-11,-2,-1],\"dy0\":0,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-10}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-11, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-10, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, Yes], [0, -11, -2, -2, 0, Yes], [1, -14, -4, -2, 0, Yes]], -1, -10, 0, -10, 0, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, Yes], [0, -11, -2, -2, 0, Yes], [1, -14, -4, -2, 0, Yes]], -1, -10, 0, -10, 0, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0078","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"508eed1213d43f1c678f205e44fb5c1f8303f933c357b7dbf77946b32cdaf02b","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214078,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-10,-1,0],\"dy0\":-1,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-10}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-10, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-10, required y_prime0=-1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, Yes], [0, -10, -1, 0, 0, Yes], [1, -11, -1, 0, 0, Yes]], 0, -10, -1, -10, -1, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, Yes], [0, -10, -1, 0, 0, Yes], [1, -11, -1, 0, 0, Yes]], 0, -10, -1, -10, -1, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0079","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad1cf05e8bb19113376ee076b6ae0ce747ed0136042b5aa0b829195ab17cfa49","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214079,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-9,0,1],\"dy0\":2,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-7}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-9, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=-7, required y_prime0=2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, Yes], [0, -9, 0, 2, 0, Yes], [1, -8, 2, 2, 0, Yes]], 1, -8, 2, -7, 2, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, Yes], [0, -9, 0, 2, 0, Yes], [1, -8, 2, 2, 0, Yes]], 1, -8, 2, -7, 2, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0080","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3d5e789fc11211e8f61258ad65f8d8111e2ac82b4f3d5ef12324589ea02423e","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214080,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-8,1,2],\"dy0\":9,\"rhs\":5,\"x0\":2,\"xs\":[-1,0,1],\"y0\":2}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-8, 1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=5; input grid=[-1, 0, 1], x0=2, required y0=2, required y_prime0=9. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, No], [0, -8, 1, 4, -1, No], [1, -5, 5, 4, -1, No]], 2, 2, 9, 2, 9, Yes, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, No], [0, -8, 1, 4, -1, No], [1, -5, 5, 4, -1, No]], 2, 2, 9, 2, 9, Yes, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0081","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00993eda3662aeab6177e46a401bf351fadc0d13863eedebed05232a6c437577","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214081,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-7,2,-2],\"dy0\":9,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-19}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-7, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=-19, required y_prime0=9. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, Yes], [0, -7, 2, -4, 0, Yes], [1, -7, -2, -4, 0, Yes]], -2, -19, 10, -19, 9, Yes, No, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, Yes], [0, -7, 2, -4, 0, Yes], [1, -7, -2, -4, 0, Yes]], -2, -19, 10, -19, 9, Yes, No, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0082","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e67a2fe0933cc3359eab02029a2e25ab0cd932821c1c396985f80ebb9fd14a5","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214082,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-6,3,-1],\"dy0\":5,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-9}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-6, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-9, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, Yes], [0, -6, 3, -2, 0, Yes], [1, -4, 1, -2, 0, Yes]], -1, -10, 5, -9, 5, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, Yes], [0, -6, 3, -2, 0, Yes], [1, -4, 1, -2, 0, Yes]], -1, -10, 5, -9, 5, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0083","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0e73774877d97e6aff9a90f3acc8d50aba4f0ad420479945bac1c3138d00246","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214083,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-5,-3,0],\"dy0\":-3,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":-5}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-5, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-5, required y_prime0=-3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, Yes], [0, -5, -3, 0, 0, Yes], [1, -8, -3, 0, 0, Yes]], 0, -5, -3, -5, -3, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, Yes], [0, -5, -3, 0, 0, Yes], [1, -8, -3, 0, 0, Yes]], 0, -5, -3, -5, -3, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0084","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82b2ca39055b6667c5b34f456c72d3b0726990e1941a2edf29eec8cbaf1d7dc9","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214084,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-4,-2,1],\"dy0\":0,\"rhs\":3,\"x0\":1,\"xs\":[-1,0,1],\"y0\":-5}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-4, -2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=3; input grid=[-1, 0, 1], x0=1, required y0=-5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0085","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7785b689eedbccf32a2060b4e5ad9e1e1dc50113d03b7b8dd9282a14e6d55b3","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214085,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-3,-1,2],\"dy0\":7,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":4}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-3, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=4, required y_prime0=7. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, Yes], [0, -3, -1, 4, 0, Yes], [1, -2, 3, 4, 0, Yes]], 2, 3, 7, 4, 7, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, Yes], [0, -3, -1, 4, 0, Yes], [1, -2, 3, 4, 0, Yes]], 2, 3, 7, 4, 7, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0086","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a4e4568f5ac33ff578b3d7647fb732edd171a787b7f1fdd2427b7d6510c5d6c","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214086,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-2,0,-2],\"dy0\":7,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-10}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-2, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=-10, required y_prime0=7. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, Yes], [0, -2, 0, -4, 0, Yes], [1, -4, -4, -4, 0, Yes]], -2, -10, 8, -10, 7, Yes, No, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, Yes], [0, -2, 0, -4, 0, Yes], [1, -4, -4, -4, 0, Yes]], -2, -10, 8, -10, 7, Yes, No, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0087","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a4236558a3999883595a9e3b3630b34abcd07ecff53fa566958582a50ddd23d","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214087,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[-1,1,-1],\"dy0\":3,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":-3}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[-1, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-3, required y_prime0=3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, Yes], [0, -1, 1, -2, 0, Yes], [1, -1, -1, -2, 0, Yes]], -1, -3, 3, -3, 3, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, Yes], [0, -1, 1, -2, 0, Yes], [1, -1, -1, -2, 0, Yes]], -1, -3, 3, -3, 3, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0088","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c3782945c183ff03c2c05efd921c184412895150561ca4227d0b093d71034e4","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214088,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[0,2,0],\"dy0\":2,\"rhs\":1,\"x0\":0,\"xs\":[-1,0,1],\"y0\":1}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[0, 2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=1; input grid=[-1, 0, 1], x0=0, required y0=1, required y_prime0=2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, No], [0, 0, 2, 0, -1, No], [1, 2, 2, 0, -1, No]], 0, 0, 2, 1, 2, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, No], [0, 0, 2, 0, -1, No], [1, 2, 2, 0, -1, No]], 0, 0, 2, 1, 2, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0089","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9298245504b9a6ec12f911a77c8f54da625a2504275329f1dcdc733204a2c36a","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214089,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[1,3,1],\"dy0\":5,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":5}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[1, 3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=5, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, Yes], [0, 1, 3, 2, 0, Yes], [1, 5, 5, 2, 0, Yes]], 1, 5, 5, 5, 5, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, Yes], [0, 1, 3, 2, 0, Yes], [1, 5, 5, 2, 0, Yes]], 1, 5, 5, 5, 5, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0090","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cef5e73fa4a2ecb66e3b53d13b0028781198eeabf78d99490e984354bcf30d16","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214090,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[2,-3,2],\"dy0\":5,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":4}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[2, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=4, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, Yes], [0, 2, -3, 4, 0, Yes], [1, 1, 1, 4, 0, Yes]], 2, 4, 5, 4, 5, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, Yes], [0, 2, -3, 4, 0, Yes], [1, 1, 1, 4, 0, Yes]], 2, 4, 5, 4, 5, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0091","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a9f9595207a13bf76cca6ef0c4ebf01665e9930965bc1ef4bc9709ff7e628fe","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214091,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[3,-2,-2],\"dy0\":5,\"rhs\":-4,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":0}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[3, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=0, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, Yes], [0, 3, -2, -4, 0, Yes], [1, -1, -6, -4, 0, Yes]], -2, -1, 6, 0, 5, No, No, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, Yes], [0, 3, -2, -4, 0, Yes], [1, -1, -6, -4, 0, Yes]], -2, -1, 6, 0, 5, No, No, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0092","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d521c2c0142f9b5d5a91fd5ed8365184b11928ec627fcc8af361f0b8f2f9751a","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214092,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[4,-1,-1],\"dy0\":1,\"rhs\":-1,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":4}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[4, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-1; input grid=[-1, 0, 1], x0=-1, required y0=4, required y_prime0=1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, No], [0, 4, -1, -2, -1, No], [1, 2, -3, -2, -1, No]], -1, 4, 1, 4, 1, Yes, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, No], [0, 4, -1, -2, -1, No], [1, 2, -3, -2, -1, No]], -1, 4, 1, 4, 1, Yes, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0093","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8ce5a42d0d28e9143138c0a396d36de78f820a4b2cc5a76411b7386af69e406","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214093,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[5,0,0],\"dy0\":0,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":5}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[5, 0, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, Yes], [0, 5, 0, 0, 0, Yes], [1, 5, 0, 0, 0, Yes]], 0, 5, 0, 5, 0, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, Yes], [0, 5, 0, 0, 0, Yes], [1, 5, 0, 0, 0, Yes]], 0, 5, 0, 5, 0, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0094","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baa97b9577b762e6d050e273d307174b15682f93434ad066c6fe41658bc629ba","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214094,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[6,1,1],\"dy0\":3,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":9}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=9, required y_prime0=3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, Yes], [0, 6, 1, 2, 0, Yes], [1, 8, 3, 2, 0, Yes]], 1, 8, 3, 9, 3, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, Yes], [0, 6, 1, 2, 0, Yes], [1, 8, 3, 2, 0, Yes]], 1, 8, 3, 9, 3, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0095","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26c6f96b80afff7adeee22795ff16e0c30812422451a49443fb7fb7115dcdceb","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214095,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[7,2,2],\"dy0\":10,\"rhs\":4,\"x0\":2,\"xs\":[-1,0,1],\"y0\":19}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[7, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=19, required y_prime0=10. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, Yes], [0, 7, 2, 4, 0, Yes], [1, 11, 6, 4, 0, Yes]], 2, 19, 10, 19, 10, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, Yes], [0, 7, 2, 4, 0, Yes], [1, 11, 6, 4, 0, Yes]], 2, 19, 10, 19, 10, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0096","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271c6aaff07faf5b3546e59875037624af4e782352ec5841b57803f53ba4a752","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214096,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[8,3,-2],\"dy0\":10,\"rhs\":-3,\"x0\":-2,\"xs\":[-1,0,1],\"y0\":-6}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[8, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-3; input grid=[-1, 0, 1], x0=-2, required y0=-6, required y_prime0=10. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, No], [0, 8, 3, -4, -1, No], [1, 9, -1, -4, -1, No]], -2, -6, 11, -6, 10, Yes, No, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, No], [0, 8, 3, -4, -1, No], [1, 9, -1, -4, -1, No]], -2, -6, 11, -6, 10, Yes, No, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0097","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a8f4364b12b3be87632f28d105fad36e399d071aaa7381909867167a137eb4f","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214097,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[9,-3,-1],\"dy0\":-1,\"rhs\":-2,\"x0\":-1,\"xs\":[-1,0,1],\"y0\":12}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[9, -3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=12, required y_prime0=-1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, Yes], [0, 9, -3, -2, 0, Yes], [1, 5, -5, -2, 0, Yes]], -1, 11, -1, 12, -1, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, Yes], [0, 9, -3, -2, 0, Yes], [1, 5, -5, -2, 0, Yes]], -1, 11, -1, 12, -1, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0098","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77859665e57b4703fc8d5e659be22961f55066604ac15f2779bb991dab75f826","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214098,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[10,-2,0],\"dy0\":-2,\"rhs\":0,\"x0\":0,\"xs\":[-1,0,1],\"y0\":10}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[10, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=10, required y_prime0=-2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, Yes], [0, 10, -2, 0, 0, Yes], [1, 8, -2, 0, 0, Yes]], 0, 10, -2, 10, -2, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, Yes], [0, 10, -2, 0, 0, Yes], [1, 8, -2, 0, 0, Yes]], 0, 10, -2, 10, -2, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0099","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f24b4b2e24b324708238e8d3adbdf55cae354f6ca40faee271e8025ff7388384","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214099,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[11,-1,1],\"dy0\":1,\"rhs\":2,\"x0\":1,\"xs\":[-1,0,1],\"y0\":11}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[11, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=11, required y_prime0=1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, Yes], [0, 11, -1, 2, 0, Yes], [1, 11, 1, 2, 0, Yes]], 1, 11, 1, 11, 1, Yes, Yes, Yes)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, Yes], [0, 11, -1, 2, 0, Yes], [1, 11, 1, 2, 0, Yes]], 1, 11, 1, 11, 1, Yes, Yes, Yes)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r04-initial-value-problem-0100","batch_id":"batch-20261013-advance-20261021-r04-initial-value-problem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f976a03df5d6a8eb13431894dceabf4c9e65eb797e36e7262568a989e224d4bf","family_id":"qf-app21-ivp-second-order-candidate-grid","seed":202610214100,"semantic_template":"qf-app21-ivp-second-order-candidate-grid/{\"a\":1,\"b\":0,\"c\":0,\"coefficients\":[12,0,2],\"dy0\":8,\"rhs\":5,\"x0\":2,\"xs\":[-1,0,1],\"y0\":21}"},"fields":{"title":"A Second-Order Candidate with Two Initial Data","stem":"Use the integer polynomial candidate with ascending-power coefficients=[12, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=5; input grid=[-1, 0, 1], x0=2, required y0=21, required y_prime0=8. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).","answer":{"kind":"expression","value":"([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, No], [0, 12, 0, 4, -1, No], [1, 14, 4, 4, -1, No]], 2, 20, 8, 21, 8, No, Yes, No)","explanation":"Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, No], [0, 12, 0, 4, -1, No], [1, 14, 4, 4, -1, No]], 2, 20, 8, 21, 8, No, Yes, No)."},"skills":["A Second-Order Candidate with Two Initial Data"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}