Question 2284: Value and Slope Initial Data as Separate Conditions

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).

• G001 — Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).

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=[-2, -2, -1]

initial point=1; required y₀=-4

y′ coefficient at degree0 := mul(1, -2)

y′ coefficient at degree0 = -2

y′ coefficient at degree1 := mul(2, -1)

y′ coefficient at degree1 = -2

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

evaluate at x=1

initial y Horner multiply before degree1 := mul(-1, 1)

initial y Horner multiply before degree1 = -1

initial y Horner add degree1 := sum(-1, -2)

initial y Horner add degree1 = -3

initial y Horner multiply before degree0 := mul(-3, 1)

initial y Horner multiply before degree0 = -3

initial y Horner add degree0 := sum(-3, -2)

initial y Horner add degree0 = -5

initial y′ ascending coefficients=[-2, -2]

evaluate at x=1

initial y′ Horner multiply before degree0 := mul(-2, 1)

initial y′ Horner multiply before degree0 = -2

initial y′ Horner add degree0 := sum(-2, -2)

initial y′ Horner add degree0 = -4

y(1)=-5; required=-4

value condition=No

y′(1)=-4; required=-4

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

([-2, -2, -1], 1, -5, -4, -4, -4, No, Yes, No)

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.