Question 1931: Candidate Value and Derivative against Two Initial Conditions

Original question and givens

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?).

• G001 — 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?).

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-ordinary-differential-equation

Step-by-step derivation

candidate y ascending coefficients=[-4, 2, -2]

audit point x=-2

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

y′ coefficient at degree0 = 2

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

y′ coefficient at degree1 = -4

y ascending coefficients=[-4, 2, -2]

evaluate at x=-2

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

y Horner multiply before degree1 = 4

y Horner add degree1 := sum(4, 2)

y Horner add degree1 = 6

y Horner multiply before degree0 := mul(6, -2)

y Horner multiply before degree0 = -12

y Horner add degree0 := sum(-12, -4)

y Horner add degree0 = -16

y′ ascending coefficients=[2, -4]

evaluate at x=-2

y′ Horner multiply before degree0 := mul(-4, -2)

y′ Horner multiply before degree0 = 8

y′ Horner add degree0 := sum(8, 2)

y′ Horner add degree0 = 10

y(-2)=-16; expected=-16

value condition: Yes

y′(-2)=10; expected=10

derivative condition: Yes

both conditions=Yes

initial conditions do not by themselves certify the ODE

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 y′ Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute 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 y Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute y Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute 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

([-4, 2, -2], -2, -16, 10, -16, 10, 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.