Question 2278: Value and Slope Initial Data as Separate Conditions

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).

• G001 — Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).

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

Step-by-step derivation

candidate polynomial ascending coefficients=[-8, -1, 2]

initial point=0; required y₀=-8

y′ coefficient at degree0 := mul(1, -1)

y′ coefficient at degree0 = -1

y′ coefficient at degree1 := mul(2, 2)

y′ coefficient at degree1 = 4

initial y ascending coefficients=[-8, -1, 2]

evaluate at x=0

initial y Horner multiply before degree1 := mul(2, 0)

initial y Horner multiply before degree1 = 0

initial y Horner add degree1 := sum(0, -1)

initial y Horner add degree1 = -1

initial y Horner multiply before degree0 := mul(-1, 0)

initial y Horner multiply before degree0 = 0

initial y Horner add degree0 := sum(0, -8)

initial y Horner add degree0 = -8

initial y′ ascending coefficients=[-1, 4]

evaluate at x=0

initial y′ Horner multiply before degree0 := mul(4, 0)

initial y′ Horner multiply before degree0 = 0

initial y′ Horner add degree0 := sum(0, -1)

initial y′ Horner add degree0 = -1

y(0)=-8; required=-8

value condition=Yes

y′(0)=-1; required=-1

derivative condition=Yes

initial data agree or disagree

no differential-equation assertion from initial data alone

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute initial y′ Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute initial y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute initial y Horner multiply before degree1 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

([-8, -1, 2], 0, -8, -1, -8, -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.