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R0003-02293 · q-20261013-advance-20261021-r04-initial-value-problem-0043
Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 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. x0=0, required y0=7, required y_prime0=0. 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=[7, 0, 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. x0=0, required y0=7, required y_prime0=0. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).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.
Before
candidate polynomial ascending coefficients=[7, 0, 2]
After
initial point=0; required y₀=7
Conditions: none
Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately.
Rule: candidate_IVP_data · Uses: G001
Before
y′ coefficient at degree0 := mul(1, 0)
After
y′ coefficient at degree0 = 0
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
y′ coefficient at degree1 := mul(2, 2)
After
y′ coefficient at degree1 = 4
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
initial y ascending coefficients=[7, 0, 2]
After
evaluate at x=0
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S003
Before
initial y Horner multiply before degree1 := mul(2, 0)
After
initial y Horner multiply before degree1 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
initial y Horner add degree1 := sum(0, 0)
After
initial y Horner add degree1 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
initial y Horner multiply before degree0 := mul(0, 0)
After
initial y Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
initial y Horner add degree0 := sum(0, 7)
After
initial y Horner add degree0 = 7
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
initial y′ ascending coefficients=[0, 4]
After
evaluate at x=0
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S008
Before
initial y′ Horner multiply before degree0 := mul(4, 0)
After
initial y′ Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
initial y′ Horner add degree0 := sum(0, 0)
After
initial y′ Horner add degree0 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
y(0)=7; required=7
After
value condition=Yes
Conditions: none
Compare the actual function value at the initial point,not another grid point.
Rule: initial_value_condition · Uses: G001, S011
Before
y′(0)=0; required=0
After
derivative condition=Yes
Conditions: none
Second-order initial data also require the derivative condition;the two conditions cannot replace one another.
Rule: initial_derivative_condition · Uses: G001, S012
Before
initial data agree or disagree
After
no differential-equation assertion from initial data alone
Conditions: none
Matching initial data do not by themselves check the differential equation or prove an existence-uniqueness theorem.
Rule: initial_only_scope · Uses: G001, S013
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.([7, 0, 2], 0, 7, 0, 7, 0, Yes, Yes, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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