Question 2250: An Initial Value Check Is Not an ODE Check

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19. Give (coefficients,x0,y(x0),required y0,match?).

• G001 — Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19. Give (coefficients,x0,y(x0),required y0,match?).

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=[14, 0, 1]

initial point=2; required y₀=19

initial y ascending coefficients=[14, 0, 1]

evaluate at x=2

initial y Horner multiply before degree1 := mul(1, 2)

initial y Horner multiply before degree1 = 2

initial y Horner add degree1 := sum(2, 0)

initial y Horner add degree1 = 2

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

initial y Horner multiply before degree0 = 4

initial y Horner add degree0 := sum(4, 14)

initial y Horner add degree0 = 18

y(2)=18; required=19

value condition=No

initial data agree or disagree

no differential-equation assertion from initial data alone

Reverse checks

• 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 — Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute initial y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([14, 0, 1], 2, 18, 19, 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.