{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r02-ordinary-differential-equation-0001","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"537833219870150eb4ec8893f9667e8b0e1e5061a65c9f82d1086535c8474cd8","family_id":"qf-app20-ode-order-classification","seed":202610202001,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0], 0, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0], 0, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0002","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47930d2cd2a6ce0e7e076a0aa80411b590eab80791e0d88b67c50f8b39074c16","family_id":"qf-app20-ode-order-classification","seed":202610202002,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1], 1, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1], 1, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0003","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e86d5bfea28c777f98092d8cb29fc5f074408e87d93a590def7efa7d73c7f3c7","family_id":"qf-app20-ode-order-classification","seed":202610202003,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1], 1, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1], 1, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0004","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a037bd617c0fcc160032e9ae6cfd2de88aab35fb1bc7620d68b200c805e94670","family_id":"qf-app20-ode-order-classification","seed":202610202004,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[2]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([2], 2, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2], 2, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0005","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"437783b68b5876bfd5785b840946e9da82904eab463cf8e47f8b928768b820e7","family_id":"qf-app20-ode-order-classification","seed":202610202005,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,2]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 2], 2, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2], 2, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0006","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2afe5033a9b085b09ab36b49a24b85307cfd72d345ec49f37df35bf89ee064d3","family_id":"qf-app20-ode-order-classification","seed":202610202006,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,2]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1, 2], 2, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2], 2, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0007","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ef07b632f95b8f49f8585cc15e82c4a697e170b45c35ee52f35d67c0605be00","family_id":"qf-app20-ode-order-classification","seed":202610202007,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,2]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1, 2], 2, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2], 2, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0008","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac27e6a432eef2d015b06084231c6f35b5a9508f814c4a004cbdf8ada0042a7d","family_id":"qf-app20-ode-order-classification","seed":202610202008,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([3], 3, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3], 3, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0009","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02336e8dd88f435bbd7290df66306834ae0a19b2ac097c5aafb6cd56b2e0b3e6","family_id":"qf-app20-ode-order-classification","seed":202610202009,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 3], 3, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3], 3, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0010","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1c0d76cec5a85f6f36b2c9de4d6f175bf94ce4c4628a482772726a895946fd5","family_id":"qf-app20-ode-order-classification","seed":202610202010,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1, 3], 3, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 3], 3, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0011","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e18953e0f5c99181a52e5ce17c201fe2f4768c88a43727c049b3007c55e21949","family_id":"qf-app20-ode-order-classification","seed":202610202011,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1, 3], 3, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], 3, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0012","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b812f8e495e49398d74318f6a7717df6ef4b27e7615cbed06af86443fbb10f0a","family_id":"qf-app20-ode-order-classification","seed":202610202012,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[2,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([2, 3], 3, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 3], 3, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0013","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c50b8614c82e91d9414e8f609825b318b19770e792b2565504b3e8f1b2fcb5ef","family_id":"qf-app20-ode-order-classification","seed":202610202013,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,2,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 2, 3], 3, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], 3, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0014","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27ae42e60eaa688a29aa2afd28499380bbede3f6ab5f5efb04da9e00e47c2ece","family_id":"qf-app20-ode-order-classification","seed":202610202014,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,2,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1, 2, 3], 3, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 3], 3, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0015","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d0d6320573672e9824e80d0f73e3ba6310626b849d67cc33537b60998373b65","family_id":"qf-app20-ode-order-classification","seed":202610202015,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,2,3]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1, 2, 3], 3, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2, 3], 3, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0016","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edbb96351727cd8b5884ac5ce976577cc67e55f55e665fd676c491cbebc5916a","family_id":"qf-app20-ode-order-classification","seed":202610202016,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([4], 4, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4], 4, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0017","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f717cab31b798fe97255570f192217aad324cc1dd5c9bffcf896fe2c7c3014fe","family_id":"qf-app20-ode-order-classification","seed":202610202017,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 4], 4, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 4], 4, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0018","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04bd5a081c9749185a002ce82ce350360cedf554c10bea20b19f33da3e0be74c","family_id":"qf-app20-ode-order-classification","seed":202610202018,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1, 4], 4, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 4], 4, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0019","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f8168aec457bb73609cda242702511b4334b625ab87be8c932fa61ad0015aa1","family_id":"qf-app20-ode-order-classification","seed":202610202019,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1, 4], 4, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 4], 4, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0020","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57ffbd860080254d2eb86f2ca26a8f14922887c5b5b88318a7f2bf2a5eabe0d4","family_id":"qf-app20-ode-order-classification","seed":202610202020,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[2,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([2, 4], 4, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 4], 4, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0021","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"828178764128c792cd4e6a629a0593a58c796ea48fad3d613ac004704ac3a118","family_id":"qf-app20-ode-order-classification","seed":202610202021,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,2,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 2, 4], 4, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 4], 4, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0022","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e115d0e7d80530749d3e374c60a929a1c3c6c4196cbc991d9f51e73a6a6cfd","family_id":"qf-app20-ode-order-classification","seed":202610202022,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[1,2,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [1, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([1, 2, 4], 4, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 4], 4, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0023","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9cbf51541311f9fc65861e941757c99c6a36005118ed2f35e7343333c9ca2bf","family_id":"qf-app20-ode-order-classification","seed":202610202023,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,1,2,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 1, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 1, 2, 4], 4, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2, 4], 4, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0024","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46e70dd4302fd81b6f13df7045cc9b59453f924d0d000f272515ee16a9f147da","family_id":"qf-app20-ode-order-classification","seed":202610202024,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[3,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [3, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([3, 4], 4, No)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 4], 4, No)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0025","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20c41b0f6049b3004214e3b4abd9d5a2f85b4217997f5779faa91c67823c2d6","family_id":"qf-app20-ode-order-classification","seed":202610202025,"semantic_template":"qf-app20-ode-order-classification/{\"derivative_orders\":[0,3,4]}"},"fields":{"title":"Highest Explicit Derivative Order, Not Candidate Polynomial Degree","stem":"Derivative orders occurring in the specified relation are [0, 3, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.","answer":{"kind":"expression","value":"([0, 3, 4], 4, Yes)","explanation":"Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3, 4], 4, Yes)."},"skills":["Highest Explicit Derivative Order, Not Candidate Polynomial Degree"],"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-20261020-r02-ordinary-differential-equation-0026","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b42061f3cbbf87b61cfc1bd428b881a6661bec5388aa659e839b196e93a2013","family_id":"qf-app20-ode-first-order-residual","seed":202610202026,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-12,1,-3],\"p\":-3,\"q\":-20,\"x\":-3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-12, 1, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=-20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-12, 1, -3], -3, -42, 19, -3, -20, -87, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-12, 1, -3], -3, -42, 19, -3, -20, -87, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0027","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"103a0eaa6a9c2c940c3ae29746f477d6fe6dffa5e071a8c2e4d3668100ded953","family_id":"qf-app20-ode-first-order-residual","seed":202610202027,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-11,2,-1],\"p\":-2,\"q\":-18,\"x\":-2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-11, 2, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=-18. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-11, 2, -1], -2, -19, 6, -2, -18, -14, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-11, 2, -1], -2, -19, 6, -2, -18, -14, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0028","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89b1311965676b53ac8f4a25d6c16ac638283d21a9083e179fbc20d847c709d","family_id":"qf-app20-ode-first-order-residual","seed":202610202028,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-10,3,1],\"p\":-1,\"q\":-16,\"x\":-1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-10, 3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=-16. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-10, 3, 1], -1, -12, 1, -1, -16, 5, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-10, 3, 1], -1, -12, 1, -1, -16, 5, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0029","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a46ca74bf41524ebd6032526bf1b00e396737957fc46379509d59e97f961403","family_id":"qf-app20-ode-first-order-residual","seed":202610202029,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-9,4,3],\"p\":0,\"q\":-14,\"x\":0}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-9, 4, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=-14. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-9, 4, 3], 0, -9, 4, 0, -14, 18, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, 4, 3], 0, -9, 4, 0, -14, 18, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0030","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e21220d92b275d0cff576c9263749f8c074a0db0d37171f46cf96823c70ea4b","family_id":"qf-app20-ode-first-order-residual","seed":202610202030,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-8,5,-2],\"p\":1,\"q\":-12,\"x\":1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, 5, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=-12. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-8, 5, -2], 1, -5, 1, 1, -12, 18, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, 5, -2], 1, -5, 1, 1, -12, 18, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0031","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b600437b26d63539065174fa08409729f2d91e6dbd4c6125ec2395c2359ba99","family_id":"qf-app20-ode-first-order-residual","seed":202610202031,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-7,1,0],\"p\":2,\"q\":-10,\"x\":2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=-10. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-7, 1, 0], 2, -5, 1, 2, -10, 21, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, 1, 0], 2, -5, 1, 2, -10, 21, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0032","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc605a212bcae62cbb76a541cd954e044f070677bf6ff14c1bf6c6adade9c41","family_id":"qf-app20-ode-first-order-residual","seed":202610202032,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-6,2,2],\"p\":3,\"q\":-8,\"x\":3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=-8. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-6, 2, 2], 3, 18, 14, 3, -8, -32, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 2, 2], 3, 18, 14, 3, -8, -32, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0033","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe6813c321cd303fa0e5ecdc2d73897687d616da830b7213b51c7f0af46280e8","family_id":"qf-app20-ode-first-order-residual","seed":202610202033,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-5,3,-3],\"p\":-3,\"q\":-6,\"x\":-3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 3, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=-6. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-5, 3, -3], -3, -41, 21, -3, -6, -96, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 3, -3], -3, -41, 21, -3, -6, -96, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0034","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"994589f828a61e5f535f1176aef2b786b3e9729c5770f98f65ede879f096f051","family_id":"qf-app20-ode-first-order-residual","seed":202610202034,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-4,4,-1],\"p\":-2,\"q\":-4,\"x\":-2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 4, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=-4. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-4, 4, -1], -2, -16, 8, -2, -4, -20, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 4, -1], -2, -16, 8, -2, -4, -20, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0035","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e1bfae906df9bfdd29abb17c3fe6bb88db6356acfa1ebc24dc90d91aadd476f","family_id":"qf-app20-ode-first-order-residual","seed":202610202035,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-3,5,1],\"p\":-1,\"q\":-2,\"x\":-1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, 5, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=-2. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-3, 5, 1], -1, -7, 3, -1, -2, -2, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 5, 1], -1, -7, 3, -1, -2, -2, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0036","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a00c42bdef6483c472ee9741a02ad917230e0b2b78926252956aa1db758c190b","family_id":"qf-app20-ode-first-order-residual","seed":202610202036,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-2,1,3],\"p\":0,\"q\":0,\"x\":0}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, 1, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=0. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-2, 1, 3], 0, -2, 1, 0, 0, 1, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, 3], 0, -2, 1, 0, 0, 1, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0037","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fab1c322a09a19ed791e2d2672f0bcb93cfcc84ecf9e0163001b7b4cb70f862","family_id":"qf-app20-ode-first-order-residual","seed":202610202037,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[-1,2,-2],\"p\":1,\"q\":2,\"x\":1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, 2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=2. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([-1, 2, -2], 1, -1, -2, 1, 2, -3, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, -2], 1, -1, -2, 1, 2, -3, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0038","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00440b7d4111cf525e7747c4f9d6ecc4fcca74e7b78ea06e72f13487d2a45b3","family_id":"qf-app20-ode-first-order-residual","seed":202610202038,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[0,3,0],\"p\":2,\"q\":4,\"x\":2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[0, 3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=4. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([0, 3, 0], 2, 6, 3, 2, 4, -13, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3, 0], 2, 6, 3, 2, 4, -13, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0039","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8afee263278bcdee9dcadf4f58e733f7ecac252e919089720e44b6641aac48c4","family_id":"qf-app20-ode-first-order-residual","seed":202610202039,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[1,4,2],\"p\":3,\"q\":6,\"x\":3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 4, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=6. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([1, 4, 2], 3, 31, 16, 3, 6, -83, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 4, 2], 3, 31, 16, 3, 6, -83, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0040","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2baef29187fa74aa19afc1e880a05bf2c0b6c1a3f83b47372400b4c022a816dd","family_id":"qf-app20-ode-first-order-residual","seed":202610202040,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[2,5,-3],\"p\":-3,\"q\":8,\"x\":-3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[2, 5, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=8. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([2, 5, -3], -3, -40, 23, -3, 8, -105, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 5, -3], -3, -40, 23, -3, 8, -105, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0041","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d24162a126c3bb8ef943c2c7f0a0e4ef14915e01493f11d64596a8d017ef404","family_id":"qf-app20-ode-first-order-residual","seed":202610202041,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[3,1,-1],\"p\":-2,\"q\":10,\"x\":-2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[3, 1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=10. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([3, 1, -1], -2, -3, 5, -2, 10, -11, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, -1], -2, -3, 5, -2, 10, -11, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0042","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"814d7c3a5f3f2c7e8422c82052292a7972347e50f1e6adb38286d199e5136226","family_id":"qf-app20-ode-first-order-residual","seed":202610202042,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[4,2,1],\"p\":-1,\"q\":12,\"x\":-1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=12. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([4, 2, 1], -1, 3, 0, -1, 12, -9, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 2, 1], -1, 3, 0, -1, 12, -9, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0043","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c6d4b47ac7abe9a4167f48f7392378a3bded44b94f9d14f04cfeca1a5d88ca","family_id":"qf-app20-ode-first-order-residual","seed":202610202043,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[5,3,3],\"p\":0,\"q\":14,\"x\":0}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[5, 3, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=14. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([5, 3, 3], 0, 5, 3, 0, 14, -11, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 3, 3], 0, 5, 3, 0, 14, -11, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0044","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ba0c769e2e65552633f61e7588e8d25dd0fb687da4d8ba5c30bb0e0e8ba9f03","family_id":"qf-app20-ode-first-order-residual","seed":202610202044,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[6,4,-2],\"p\":1,\"q\":16,\"x\":1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[6, 4, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=16. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([6, 4, -2], 1, 8, 0, 1, 16, -24, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 4, -2], 1, 8, 0, 1, 16, -24, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0045","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c50f527feec8521ad8ed3c357a1e0125e5158986eff95326c57d97fe995cabd","family_id":"qf-app20-ode-first-order-residual","seed":202610202045,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[7,5,0],\"p\":2,\"q\":18,\"x\":2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[7, 5, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=18. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([7, 5, 0], 2, 17, 5, 2, 18, -47, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, 5, 0], 2, 17, 5, 2, 18, -47, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0046","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e234b806e4ac12bda982d63d56b170db381b51c4c56940ef7a083db41066b05","family_id":"qf-app20-ode-first-order-residual","seed":202610202046,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([8, 1, 2], 3, 29, 13, 3, 20, -94, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, 1, 2], 3, 29, 13, 3, 20, -94, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0047","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12ad4cb78e49c91fc5e3f506d9a6c2d91e31411b35a2284664533018a066c864","family_id":"qf-app20-ode-first-order-residual","seed":202610202047,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[9,2,-3],\"p\":-3,\"q\":22,\"x\":-3}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 2, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=22. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([9, 2, -3], -3, -24, 20, -3, 22, -74, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 2, -3], -3, -24, 20, -3, 22, -74, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0048","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c24b51abb9643095db56c03b6768ea3a53024be69162970f4f60d9f29478e36e","family_id":"qf-app20-ode-first-order-residual","seed":202610202048,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[10,3,-1],\"p\":-2,\"q\":24,\"x\":-2}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 3, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=24. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([10, 3, -1], -2, 0, 7, -2, 24, -17, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 3, -1], -2, 0, 7, -2, 24, -17, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0049","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef3d035ca785a975ae3c2953cf30528e32d038444ed937eb53d919f0f33ef57a","family_id":"qf-app20-ode-first-order-residual","seed":202610202049,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[11,4,1],\"p\":-1,\"q\":26,\"x\":-1}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 4, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=26. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([11, 4, 1], -1, 8, 2, -1, 26, -16, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 4, 1], -1, 8, 2, -1, 26, -16, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0050","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1c4df853b07faae577af486e7fa65640c9f1047e1b33723d2ef11c3f432c72f","family_id":"qf-app20-ode-first-order-residual","seed":202610202050,"semantic_template":"qf-app20-ode-first-order-residual/{\"coefficients\":[12,5,3],\"p\":0,\"q\":28,\"x\":0}"},"fields":{"title":"One-Point Residual for a First-Order Relation","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[12, 5, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=28. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).","answer":{"kind":"expression","value":"([12, 5, 3], 0, 12, 5, 0, 28, -23, No)","explanation":"Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, 5, 3], 0, 12, 5, 0, 28, -23, No)."},"skills":["One-Point Residual for a First-Order Relation"],"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-20261020-r02-ordinary-differential-equation-0051","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ebaf84c9a2301c27185ff1396946c48193f3d949c7725f3365a00969da6d0bd","family_id":"qf-app20-ode-second-order-residual","seed":202610202051,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":-3,\"c\":-4,\"coefficients\":[-8,-2,-3,0],\"rhs\":-20,\"x\":-2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, -2, -3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-2, check a*y''+b*y'+c*y=rhs with a=1≠0, b=-3, c=-4, rhs=-20. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0052","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13e6825a7949c326af7e0c8f804bf976815f4b2d4e8ca409570a17fee384a181","family_id":"qf-app20-ode-second-order-residual","seed":202610202052,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-2,\"c\":-3,\"coefficients\":[-7,-1,0,1],\"rhs\":-17,\"x\":-1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, -1, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-2, c=-3, rhs=-17. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0053","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f9ec458a30a911c65a4164bca4e529c47baa016ba1e6d14fd245fb4a103e913","family_id":"qf-app20-ode-second-order-residual","seed":202610202053,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-1,\"c\":-2,\"coefficients\":[-6,0,3,2],\"rhs\":-14,\"x\":0}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 0, 3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-1, c=-2, rhs=-14. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0054","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86bdebc89c3e1ab5e4a95e8b9c25085d02b961f9385ec40e1e569bb17b75b0a9","family_id":"qf-app20-ode-second-order-residual","seed":202610202054,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":0,\"c\":-1,\"coefficients\":[-5,1,-1,0],\"rhs\":-11,\"x\":1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 1, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=0, c=-1, rhs=-11. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0055","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3a4c87ed7d3d3ec8c7f2a5cb611af05ebb92d9b5bf02f83436f05e889462c18","family_id":"qf-app20-ode-second-order-residual","seed":202610202055,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":1,\"c\":0,\"coefficients\":[-4,2,2,1],\"rhs\":-8,\"x\":2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 2, 2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=1, c=0, rhs=-8. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0056","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6717d2705f3bc8fd6c56a1bbecafdee40dc4f8272dc06773b6853b20250b038d","family_id":"qf-app20-ode-second-order-residual","seed":202610202056,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":2,\"c\":1,\"coefficients\":[-3,-2,-2,2],\"rhs\":-5,\"x\":-2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, -2, -2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-2, check a*y''+b*y'+c*y=rhs with a=1≠0, b=2, c=1, rhs=-5. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0057","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2d0af8d6bed708c2752570550007a91e850b063809e6d8f75aa0ac3575205c6","family_id":"qf-app20-ode-second-order-residual","seed":202610202057,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":3,\"c\":2,\"coefficients\":[-2,-1,1,0],\"rhs\":-2,\"x\":-1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, -1, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=3, c=2, rhs=-2. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0058","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6e1c52a2a4bd8df704af1f6fb9e7f4ff3a97a23cef896de194e488fd8889b33","family_id":"qf-app20-ode-second-order-residual","seed":202610202058,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-3,\"c\":3,\"coefficients\":[-1,0,-3,1],\"rhs\":1,\"x\":0}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, 0, -3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-3, c=3, rhs=1. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0059","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad50cbbdf1dab402d6937e3ac52aeae4171c90b3fcc850ac1a85ecaf370d81c","family_id":"qf-app20-ode-second-order-residual","seed":202610202059,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":-2,\"c\":4,\"coefficients\":[0,1,0,2],\"rhs\":4,\"x\":1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[0, 1, 0, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=-2, c=4, rhs=4. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0060","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f05c845debf18250795d386a74a28096ed701596ee7f124081bc9afda62ab488","family_id":"qf-app20-ode-second-order-residual","seed":202610202060,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":-1,\"c\":-4,\"coefficients\":[1,2,3,0],\"rhs\":7,\"x\":2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 2, 3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=-1, c=-4, rhs=7. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0061","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d6fadc1ba032106a6f47d6fe1a9e89ecf5c36d5ea0600cc15dc1ceae4efa8f","family_id":"qf-app20-ode-second-order-residual","seed":202610202061,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":0,\"c\":-3,\"coefficients\":[2,-2,-1,1],\"rhs\":10,\"x\":-2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[2, -2, -1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-2, check a*y''+b*y'+c*y=rhs with a=1≠0, b=0, c=-3, rhs=10. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0062","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64f94d2e37234fe262e6cc37e702fb7de5655554a479c1c296e2b9bdd7f20ead","family_id":"qf-app20-ode-second-order-residual","seed":202610202062,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":1,\"c\":-2,\"coefficients\":[3,-1,2,2],\"rhs\":13,\"x\":-1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[3, -1, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=1, c=-2, rhs=13. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0063","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5d82096b3f7942b3e93a30ac6c5e4015c92a5ae6f4614b3a86140e8726c50a0","family_id":"qf-app20-ode-second-order-residual","seed":202610202063,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":2,\"c\":-1,\"coefficients\":[4,0,-2,0],\"rhs\":16,\"x\":0}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 0, -2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=2, c=-1, rhs=16. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0064","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc7bbfa69cbdacbc4027041581091ec12179340221c63b2fcac6786bedda6be4","family_id":"qf-app20-ode-second-order-residual","seed":202610202064,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":3,\"c\":0,\"coefficients\":[5,1,1,1],\"rhs\":19,\"x\":1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[5, 1, 1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=3, c=0, rhs=19. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0065","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e27ca31bf033eefdeb3213b547e99a4ec0fbfa8fe6642025f36a2a074ab9054a","family_id":"qf-app20-ode-second-order-residual","seed":202610202065,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":-3,\"c\":1,\"coefficients\":[6,2,-3,2],\"rhs\":22,\"x\":2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[6, 2, -3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=-3, c=1, rhs=22. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0066","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d78a6c9d84755b139ddfe2f25975cbd7bab95f032d11dfb8241f2c0b99338c","family_id":"qf-app20-ode-second-order-residual","seed":202610202066,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":-2,\"c\":2,\"coefficients\":[7,-2,0,0],\"rhs\":25,\"x\":-2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[7, -2, 0, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-2, check a*y''+b*y'+c*y=rhs with a=1≠0, b=-2, c=2, rhs=25. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0067","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef7ac26eb02d9901947d063139f83ac8cd78c8eff218fbad775923e9aae2bf6","family_id":"qf-app20-ode-second-order-residual","seed":202610202067,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-1,\"c\":3,\"coefficients\":[8,-1,3,1],\"rhs\":28,\"x\":-1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[8, -1, 3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-1, c=3, rhs=28. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0068","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64ac36ce51d9fbf89f632f6c61de0f584fbbc11e3913316e358202f17868865f","family_id":"qf-app20-ode-second-order-residual","seed":202610202068,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":0,\"c\":4,\"coefficients\":[9,0,-1,2],\"rhs\":31,\"x\":0}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 0, -1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=0, c=4, rhs=31. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0069","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98abebf2306facb398b8f9ff1f65d972960265c8cbab8375db02575ce69813b3","family_id":"qf-app20-ode-second-order-residual","seed":202610202069,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":1,\"c\":-4,\"coefficients\":[10,1,2,0],\"rhs\":34,\"x\":1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 1, 2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=1, c=-4, rhs=34. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0070","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c877bc6c2b7e93577c2a2916dbb69320c89c1cafa99607a0793be0291b5928e","family_id":"qf-app20-ode-second-order-residual","seed":202610202070,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":2,\"c\":-3,\"coefficients\":[11,2,-2,1],\"rhs\":37,\"x\":2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 2, -2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=2, c=-3, rhs=37. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0071","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d48c001b31902760a2ba00fbc167cb1c075f8776d9e2fa784d75aa0cef81e12","family_id":"qf-app20-ode-second-order-residual","seed":202610202071,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":1,\"b\":3,\"c\":-2,\"coefficients\":[12,-2,1,2],\"rhs\":40,\"x\":-2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[12, -2, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-2, check a*y''+b*y'+c*y=rhs with a=1≠0, b=3, c=-2, rhs=40. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0072","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9d99b1a46519e1aa076edd13fba5f4e51f787ce87cbf7fd61864a4bb984998c","family_id":"qf-app20-ode-second-order-residual","seed":202610202072,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":2,\"b\":-3,\"c\":-1,\"coefficients\":[13,-1,-3,0],\"rhs\":43,\"x\":-1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[13, -1, -3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-3, c=-1, rhs=43. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0073","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb67c54477b13688d80c0d3693f6c67314b397d63cb92fe4d8ea1b0a35e8316e","family_id":"qf-app20-ode-second-order-residual","seed":202610202073,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":3,\"b\":-2,\"c\":0,\"coefficients\":[14,0,0,1],\"rhs\":46,\"x\":0}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[14, 0, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-2, c=0, rhs=46. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0074","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c3a15646e8af603f28b0b812b1e6e79f84266b75c98ea49060611c3b0a1d845","family_id":"qf-app20-ode-second-order-residual","seed":202610202074,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":4,\"b\":-1,\"c\":1,\"coefficients\":[15,1,3,2],\"rhs\":49,\"x\":1}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[15, 1, 3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=-1, c=1, rhs=49. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0075","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e683a259a5e23afe61cd7304a15985e7463850798dcf5f466e06e803da42895c","family_id":"qf-app20-ode-second-order-residual","seed":202610202075,"semantic_template":"qf-app20-ode-second-order-residual/{\"a\":5,\"b\":0,\"c\":2,\"coefficients\":[16,2,-1,0],\"rhs\":52,\"x\":2}"},"fields":{"title":"One-Point Residual with a Nonzero Second-Order Coefficient","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[16, 2, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=0, c=2, rhs=52. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).","answer":{"kind":"expression","value":"([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, No)","explanation":"Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, No)."},"skills":["One-Point Residual with a Nonzero Second-Order Coefficient"],"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-20261020-r02-ordinary-differential-equation-0076","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e47dc590930fb3261772361792a85a281db3a85bf28e5fedeea1e7fe88bec273","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202076,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-9,-3,-2],\"dy0\":4,\"x0\":-2,\"y0\":-10}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-9, -3, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-10, required y'0=4. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-9, -3, -2], -2, -11, 5, -10, 4, No, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, -3, -2], -2, -11, 5, -10, 4, No, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0077","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2a28d1922d470bf31b1b861de4f41ad83b4c82814ac1736c75bd4a11d6ff64c","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202077,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-8,-2,0],\"dy0\":-2,\"x0\":-1,\"y0\":-6}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, -2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=-6, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-8, -2, 0], -1, -6, -2, -6, -2, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, 0], -1, -6, -2, -6, -2, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0078","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c1e222e42edaafab279612c7ca9fe264f41e7e5b66b72e6b35d5a7f307357e3","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202078,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-7,-1,2],\"dy0\":-1,\"x0\":0,\"y0\":-7}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, -1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=-7, required y'0=-1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-7, -1, 2], 0, -7, -1, -7, -1, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, -1, 2], 0, -7, -1, -7, -1, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0079","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba090ffeb780f82b0007cccfdc6bfb6e3ea23624a5d202dfd6121bd4a814dee","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202079,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-6,0,-1],\"dy0\":-2,\"x0\":1,\"y0\":-6}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 0, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=-6, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-6, 0, -1], 1, -7, -2, -6, -2, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 0, -1], 1, -7, -2, -6, -2, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0080","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ece077ca8ebb2399ad8ec4c1695fa5980cbee49190b9b5417ef851a5ff51defe","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202080,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-5,1,1],\"dy0\":4,\"x0\":2,\"y0\":1}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=1, required y'0=4. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-5, 1, 1], 2, 1, 5, 1, 4, Yes, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 1, 1], 2, 1, 5, 1, 4, Yes, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0081","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9339d47f2f6aa82b9070509e6c4ff6e14a7cd9b2932a844c6ae6c623e42b1f06","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202081,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-4,2,-2],\"dy0\":10,\"x0\":-2,\"y0\":-16}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-16, required y'0=10. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-4, 2, -2], -2, -16, 10, -16, 10, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 2, -2], -2, -16, 10, -16, 10, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0082","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce5e5326c2fe383db10cf2b9227a0bde655f0ea17d0acd65dcca2742e3b2c8b","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202082,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-3,3,0],\"dy0\":3,\"x0\":-1,\"y0\":-5}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, 3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=-5, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-3, 3, 0], -1, -6, 3, -5, 3, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 3, 0], -1, -6, 3, -5, 3, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0083","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44cc0067dff0fdcae2baa6543970cb20e840c574489b8f76e5a4a20915a568e","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202083,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-2,-3,2],\"dy0\":-3,\"x0\":0,\"y0\":-2}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, -3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=-2, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-2, -3, 2], 0, -2, -3, -2, -3, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -3, 2], 0, -2, -3, -2, -3, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0084","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13f33afc90e78d365722ae2912a25956dc43bc137597a385d28dd949e9c9c64c","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202084,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[-1,-2,-1],\"dy0\":-5,\"x0\":1,\"y0\":-4}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, -2, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=-4, required y'0=-5. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([-1, -2, -1], 1, -4, -4, -4, -5, Yes, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -2, -1], 1, -4, -4, -4, -5, Yes, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0085","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ebda2d1c8c1992db6b3bbf6e9f1cfb6af23df9f593bac2ce27f00a96577aebe","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202085,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[0,-1,1],\"dy0\":3,\"x0\":2,\"y0\":3}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[0, -1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=3, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([0, -1, 1], 2, 2, 3, 3, 3, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 1], 2, 2, 3, 3, 3, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0086","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f0c3b650c8ec32adab03c5f30aebb12ccffbb5cf859cf29d234572ef72ebe08","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202086,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[1,0,-2],\"dy0\":8,\"x0\":-2,\"y0\":-7}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 0, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-7, required y'0=8. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([1, 0, -2], -2, -7, 8, -7, 8, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], -2, -7, 8, -7, 8, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0087","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"234ede7bcee0606b95b8f66e67a8001d87d53430b5f424def23419892227726d","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202087,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[2,1,0],\"dy0\":1,\"x0\":-1,\"y0\":1}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[2, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=1, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([2, 1, 0], -1, 1, 1, 1, 1, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], -1, 1, 1, 1, 1, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0088","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abfca014b7dfb6880908910cd55ecad80a4785df685e403b163ed33e08778b68","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202088,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[3,2,2],\"dy0\":1,\"x0\":0,\"y0\":4}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[3, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=4, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([3, 2, 2], 0, 3, 2, 4, 1, No, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 2, 2], 0, 3, 2, 4, 1, No, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0089","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"998045f5ff7bb73dc6607f903f2c2e88b1d07824310a59e9ee642bc7bc940bd9","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202089,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[4,3,-1],\"dy0\":1,\"x0\":1,\"y0\":6}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 3, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=6, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([4, 3, -1], 1, 6, 1, 6, 1, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 3, -1], 1, 6, 1, 6, 1, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0090","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"314919db3b740926f8202bbde4a5d893c1de6ce610bc671239f71996e1902f4b","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202090,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[5,-3,1],\"dy0\":1,\"x0\":2,\"y0\":3}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[5, -3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=3, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([5, -3, 1], 2, 3, 1, 3, 1, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, -3, 1], 2, 3, 1, 3, 1, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0091","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31b916baad43035c93d37a25f6182052ea2a5ddb9267597854227588b25b9783","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202091,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[6,-2,-2],\"dy0\":6,\"x0\":-2,\"y0\":3}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[6, -2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=3, required y'0=6. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([6, -2, -2], -2, 2, 6, 3, 6, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, -2, -2], -2, 2, 6, 3, 6, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0092","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b96fee7f2d1d7b7c8ee1dc53dadaa1880e464a5713b752ce2ca7a2220bc846ec","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202092,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[7,-1,0],\"dy0\":-2,\"x0\":-1,\"y0\":8}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[7, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=8, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([7, -1, 0], -1, 8, -1, 8, -2, Yes, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -1, 0], -1, 8, -1, 8, -2, Yes, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0093","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3bb50d06f966e2c9c9472d2e08e14d8f979fc9b30a9907d9baa5b71def3b687","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202093,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[8,0,2],\"dy0\":0,\"x0\":0,\"y0\":8}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 0, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=8, required y'0=0. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([8, 0, 2], 0, 8, 0, 8, 0, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, 0, 2], 0, 8, 0, 8, 0, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0094","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f7045d87bf489a61b38d98598af50434198e5d403a0c52275324657df7bb05","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202094,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[9,1,-1],\"dy0\":-1,\"x0\":1,\"y0\":10}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=10, required y'0=-1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([9, 1, -1], 1, 9, -1, 10, -1, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 1, -1], 1, 9, -1, 10, -1, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0095","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"153f3417d758d1be998de73894b9ea613c5595813306763cc829320fe0f87cb8","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202095,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[10,2,1],\"dy0\":6,\"x0\":2,\"y0\":18}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=18, required y'0=6. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([10, 2, 1], 2, 18, 6, 18, 6, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 2, 1], 2, 18, 6, 18, 6, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0096","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343e4075ea93e613feb70fc7bd09c095fb0515ba804ec578e5c604f3826aac66","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202096,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[11,3,-2],\"dy0\":10,\"x0\":-2,\"y0\":-3}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 3, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-3, required y'0=10. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([11, 3, -2], -2, -3, 11, -3, 10, Yes, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 3, -2], -2, -3, 11, -3, 10, Yes, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0097","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c313f295dbe05b940d903f92a9a1684555ca50b177768471afaea4813dfee024","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202097,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[12,-3,0],\"dy0\":-3,\"x0\":-1,\"y0\":16}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[12, -3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=16, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([12, -3, 0], -1, 15, -3, 16, -3, No, Yes, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, -3, 0], -1, 15, -3, 16, -3, No, Yes, No)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0098","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfe8bc29182410e3805875c7cdcdd79868ef97dbbbfd98d708f0e4d93cfd6e75","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202098,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[13,-2,2],\"dy0\":-2,\"x0\":0,\"y0\":13}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[13, -2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=13, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([13, -2, 2], 0, 13, -2, 13, -2, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -2, 2], 0, 13, -2, 13, -2, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0099","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65985c9023ecf516b1834b482e8aa18f22668b7476ca4a6862fa0bc11cd95b70","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202099,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[14,-1,-1],\"dy0\":-3,\"x0\":1,\"y0\":12}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[14, -1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=12, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([14, -1, -1], 1, 12, -3, 12, -3, Yes, Yes, Yes)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, -1, -1], 1, 12, -3, 12, -3, Yes, Yes, Yes)."},"skills":["Candidate Value and Derivative against Two Initial 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-20261020-r02-ordinary-differential-equation-0100","batch_id":"batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a70ee714738a4d56587da4d32eb7b91ff7d6f8f76f4947dbd8816efeca8ab23","family_id":"qf-app20-ode-initial-condition-audit","seed":202610202100,"semantic_template":"qf-app20-ode-initial-condition-audit/{\"coefficients\":[15,0,1],\"dy0\":3,\"x0\":2,\"y0\":20}"},"fields":{"title":"Candidate Value and Derivative against Two Initial Conditions","stem":"The integer-coefficient polynomial candidate has ascending-power coefficients=[15, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=20, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).","answer":{"kind":"expression","value":"([15, 0, 1], 2, 19, 4, 20, 3, No, No, No)","explanation":"Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([15, 0, 1], 2, 19, 4, 20, 3, No, No, No)."},"skills":["Candidate Value and Derivative against Two Initial 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."}}