Question 2209: A First-Order Candidate: Grid Residuals and Initial Value

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[-4, -2]. 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. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).

• G001 — Use the integer polynomial candidate with ascending-power coefficients=[-4, -2]. 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. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial 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=[-4, -2]

initial point=1; required y₀=-6

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

y′ coefficient at degree0 = -2

initial y ascending coefficients=[-4, -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, -4)

initial y Horner add degree0 = -6

y(1)=-6; required=-6

value condition=Yes

[-1, 0, 1]

distinct ordered audit grid=[-1, 0, 1]

y at-1 ascending coefficients=[-4, -2]

evaluate at x=-1

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

y at-1 Horner multiply before degree0 = 2

y at-1 Horner add degree0 := sum(2, -4)

y at-1 Horner add degree0 = -2

y′ at-1 ascending coefficients=[-2]

evaluate at x=-1

p·y at-1 := mul(0, -2)

p·y at-1 = 0

y′−p·y at-1 := sub(-2, 0)

y′−p·y at-1 = -2

residual at-1 := sub(-2, -1)

residual at-1 = -1

residual at-1=-1

point equation match=No

y at0 ascending coefficients=[-4, -2]

evaluate at x=0

y at0 Horner multiply before degree0 := mul(-2, 0)

y at0 Horner multiply before degree0 = 0

y at0 Horner add degree0 := sum(0, -4)

y at0 Horner add degree0 = -4

y′ at0 ascending coefficients=[-2]

evaluate at x=0

p·y at0 := mul(0, -4)

p·y at0 = 0

y′−p·y at0 := sub(-2, 0)

y′−p·y at0 = -2

residual at0 := sub(-2, -1)

residual at0 = -1

residual at0=-1

point equation match=No

y at1 ascending coefficients=[-4, -2]

evaluate at x=1

y at1 Horner multiply before degree0 := mul(-2, 1)

y at1 Horner multiply before degree0 = -2

y at1 Horner add degree0 := sum(-2, -4)

y at1 Horner add degree0 = -6

y′ at1 ascending coefficients=[-2]

evaluate at x=1

p·y at1 := mul(0, -6)

p·y at1 = 0

y′−p·y at1 := sub(-2, 0)

y′−p·y at1 = -2

residual at1 := sub(-2, -1)

residual at1 = -1

residual at1=-1

point equation match=No

all listed grid points pass=No; initial conditions pass=Yes

combined finite check=No

finite grid agreement alone is not a general interval certificate

a polynomial coefficient/degree certificate is a separate possible strengthening

Reverse checks

• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute y′−p·y at1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute p·y at1 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute residual at0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute y′−p·y at0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute p·y at0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• 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 residual at-1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute y′−p·y at-1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute p·y at-1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, No], [0, -4, -2, -1, No], [1, -6, -2, -1, No]], 1, -6, -6, 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.