Question 2285: Value and Slope Initial Data as Separate Conditions

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 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=1, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).

• G001 — Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 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=1, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,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-initial-value-problem

Step-by-step derivation

candidate polynomial ascending coefficients=[-1, -1, 1]

initial point=2; required y₀=1

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

y′ coefficient at degree0 = -1

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

y′ coefficient at degree1 = 2

initial y ascending coefficients=[-1, -1, 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, -1)

initial y Horner add degree1 = 1

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

initial y Horner multiply before degree0 = 2

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

initial y Horner add degree0 = 1

initial y′ ascending coefficients=[-1, 2]

evaluate at x=2

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

initial y′ Horner multiply before degree0 = 4

initial y′ Horner add degree0 := sum(4, -1)

initial y′ Horner add degree0 = 3

y(2)=1; required=1

value condition=Yes

y′(2)=3; required=2

derivative condition=No

initial data agree or disagree

no differential-equation assertion from initial data alone

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

([-1, -1, 1], 2, 1, 3, 1, 2, Yes, No, 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.