Question 2000: One-Point Residual with a Nonzero Second-Order Coefficient

Original question and givens

The integer-coefficient polynomial candidate has ascending-power coefficients=[16, 2, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=0, c=2, rhs=52. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

• G001 — The integer-coefficient polynomial candidate has ascending-power coefficients=[16, 2, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=2, check a*y''+b*y'+c*y=rhs with a=5≠0, b=0, c=2, rhs=52. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

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=[16, 2, -1, 0]

audit point x=2

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

y′ coefficient at degree0 = 2

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

y′ coefficient at degree1 = -2

y′ coefficient at degree2 := mul(3, 0)

y′ coefficient at degree2 = 0

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

y″ coefficient at degree0 = -2

y″ coefficient at degree1 := mul(2, 0)

y″ coefficient at degree1 = 0

y ascending coefficients=[16, 2, -1, 0]

evaluate at x=2

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

y Horner multiply before degree1 = -2

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

y Horner add degree1 = 0

y Horner multiply before degree0 := mul(0, 2)

y Horner multiply before degree0 = 0

y Horner add degree0 := sum(0, 16)

y Horner add degree0 = 16

y′ ascending coefficients=[2, -2, 0]

evaluate at x=2

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

y′ Horner multiply before degree0 = -4

y′ Horner add degree0 := sum(-4, 2)

y′ Horner add degree0 = -2

y″ ascending coefficients=[-2, 0]

evaluate at x=2

a·y″ := mul(5, -2)

a·y″ = -10

b·y′ := mul(0, -2)

b·y′ = 0

c·y := mul(2, 16)

c·y = 32

a·y″+b·y′+c·y := sum(-10, 0, 32)

a·y″+b·y′+c·y = 22

second-order residual := sub(22, 52)

second-order residual = -30

residual at x=2 is -30

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

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute second-order residual with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute a·y″+b·y′+c·y with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute c·y with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute b·y′ with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute a·y″ with exact rational arithmetic, not rounded decimals.
• 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′ Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• 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 — Recompute y Horner add degree1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
• 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 y″ coefficient at degree1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute y″ coefficient at degree0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute y′ coefficient at degree2 with exact rational arithmetic, not rounded decimals.
• 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

([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, 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.