Question 1921: One-Point Residual for a First-Order Relation

Original question and givens

The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

• G001 — The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

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=[8, 1, 2]

audit point x=3

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

y′ coefficient at degree0 = 1

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

y′ coefficient at degree1 = 4

y ascending coefficients=[8, 1, 2]

evaluate at x=3

y Horner multiply before degree1 := mul(2, 3)

y Horner multiply before degree1 = 6

y Horner add degree1 := sum(6, 1)

y Horner add degree1 = 7

y Horner multiply before degree0 := mul(7, 3)

y Horner multiply before degree0 = 21

y Horner add degree0 := sum(21, 8)

y Horner add degree0 = 29

y′ ascending coefficients=[1, 4]

evaluate at x=3

y′ Horner multiply before degree0 := mul(4, 3)

y′ Horner multiply before degree0 = 12

y′ Horner add degree0 := sum(12, 1)

y′ Horner add degree0 = 13

p·y := mul(3, 29)

p·y = 87

y′−p·y := sub(13, 87)

y′−p·y = -74

y′−p·y−q := sub(-74, 20)

y′−p·y−q = -94

residual at x=3 is -94

equation holds at this point: No

global solution would require the residual identity on the declared domain

no automatic existence,uniqueness or global-solution claim

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute y′−p·y−q with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute y′−p·y with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute p·y with exact rational arithmetic, not rounded decimals.
• 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

([8, 1, 2], 3, 29, 13, 3, 20, -94, 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.