# Question 2287: Value and Slope Initial Data as Separate Conditions [Return to catalogue page 12](https://ocme.evemisslab.com/en/questions/page/12/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r04-initial-value-problem-foundations/#q-20261013-advance-20261021-r04-initial-value-problem-0037) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0003-02287 · q-20261013-advance-20261021-r04-initial-value-problem-0037 ## Original question and givens Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?). - G001 — Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-initial-value-problem](https://ocme.evemisslab.com/en/mko/mko-initial-value-problem/) ## Step-by-step derivation S001 · impliesBefore candidate polynomial ascending coefficients=[1, 1, 0] After initial point=-1; required y₀=0 Conditions: none Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately. Step identifiers and dependenciesRule: candidate_IVP_data · Uses: G001 S002 · calculationBefore y′ coefficient at degree0 := mul(1, 1) After y′ coefficient at degree0 = 1 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore y′ coefficient at degree1 := mul(2, 0) After y′ coefficient at degree1 = 0 Conditions: none Multiply by the original degree and lower it by1;the constant term differentiates to0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore initial y ascending coefficients=[1, 1, 0] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S003 S005 · calculationBefore initial y Horner multiply before degree0 := mul(1, -1) After initial y Horner multiply before degree0 = -1 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore initial y Horner add degree0 := sum(-1, 1) After initial y Horner add degree0 = 0 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore initial y′ ascending coefficients=[1, 0] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S006 S008 · impliesBefore y(-1)=0; required=0 After value condition=Yes Conditions: none Compare the actual function value at the initial point,not another grid point. Step identifiers and dependenciesRule: initial_value_condition · Uses: G001, S007 S009 · impliesBefore y′(-1)=1; required=1 After derivative condition=Yes Conditions: none Second-order initial data also require the derivative condition;the two conditions cannot replace one another. Step identifiers and dependenciesRule: initial_derivative_condition · Uses: G001, S008 S010 · impliesBefore initial data agree or disagree After no differential-equation assertion from initial data alone Conditions: none Matching initial data do not by themselves check the differential equation or prove an existence-uniqueness theorem. Step identifiers and dependenciesRule: initial_only_scope · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute y′ coefficient at degree1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute y′ coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([1, 1, 0], -1, 0, 1, 0, 1, Yes, Yes, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2e274dd81e7e71ae1c4e06a244edaad3655e00b1b2338c4bc9e7a90138282c8b - candidate record SHA-256: 5bf778017559a9986679c4229aa86a04689eb7343ee75cfac6e3aae11ea22652 - semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f - question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r04-initial-value-problem-0037.json)