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R0002-01996 · q-20261013-advance-20261020-r02-ordinary-differential-equation-0071
The integer-coefficient polynomial candidate has ascending-power coefficients=[12, -2, 1, 2]. 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=1≠0, b=3, c=-2, rhs=40. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).
G001 — The integer-coefficient polynomial candidate has ascending-power coefficients=[12, -2, 1, 2]. 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=1≠0, b=3, c=-2, rhs=40. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).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 y ascending coefficients=[12, -2, 1, 2]
After
audit point x=-2
Conditions: none
Retain the original coefficient list,including top zero terms,and audit only the specified polynomial candidate and point.
Rule: polynomial_candidate_domain · Uses: G001
Before
y′ coefficient at degree0 := mul(1, -2)
After
y′ coefficient at degree0 = -2
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, 1)
After
y′ coefficient at degree1 = 2
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
y′ coefficient at degree2 := mul(3, 2)
After
y′ coefficient at degree2 = 6
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
y″ coefficient at degree0 := mul(1, 2)
After
y″ coefficient at degree0 = 2
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
y″ coefficient at degree1 := mul(2, 6)
After
y″ coefficient at degree1 = 12
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
y ascending coefficients=[12, -2, 1, 2]
After
evaluate at x=-2
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, S006
Before
y Horner multiply before degree2 := mul(2, -2)
After
y Horner multiply before degree2 = -4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
y Horner add degree2 := sum(-4, 1)
After
y Horner add degree2 = -3
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
y Horner multiply before degree1 := mul(-3, -2)
After
y Horner multiply before degree1 = 6
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
y Horner add degree1 := sum(6, -2)
After
y Horner add degree1 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
y Horner multiply before degree0 := mul(4, -2)
After
y Horner multiply before degree0 = -8
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
y Horner add degree0 := sum(-8, 12)
After
y Horner add degree0 = 4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
y′ ascending coefficients=[-2, 2, 6]
After
evaluate at x=-2
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, S013
Before
y′ Horner multiply before degree1 := mul(6, -2)
After
y′ Horner multiply before degree1 = -12
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
y′ Horner add degree1 := sum(-12, 2)
After
y′ Horner add degree1 = -10
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
y′ Horner multiply before degree0 := mul(-10, -2)
After
y′ Horner multiply before degree0 = 20
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
y′ Horner add degree0 := sum(20, -2)
After
y′ Horner add degree0 = 18
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
y″ ascending coefficients=[2, 12]
After
evaluate at x=-2
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, S018
Before
y″ Horner multiply before degree0 := mul(12, -2)
After
y″ Horner multiply before degree0 = -24
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
y″ Horner add degree0 := sum(-24, 2)
After
y″ Horner add degree0 = -22
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
a·y″ := mul(1, -22)
After
a·y″ = -22
Conditions: none
Multiply each derivative by its own original equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
b·y′ := mul(3, 18)
After
b·y′ = 54
Conditions: none
Multiply each derivative by its own original equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
c·y := mul(-2, 4)
After
c·y = -8
Conditions: none
Multiply each derivative by its own original equation coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
a·y″+b·y′+c·y := sum(-22, 54, -8)
After
a·y″+b·y′+c·y = 24
Conditions: none
Sum all three terms before comparing with rhs.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
second-order residual := sub(24, 40)
After
second-order residual = -16
Conditions: none
Compute the residual with the same left-minus-right sign convention.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
residual at x=-2 is -16
After
equation holds at this point: No
Conditions: none
This decides the equation at this point only;a zero residual at one point is not a solution on all reals.
Rule: point_residual_verdict · Uses: G001, S026
Before
global solution would require the residual identity on the declared domain
After
no automatic existence,uniqueness or global-solution claim
Conditions: none
Equation order depends on the nonzero leading derivative coefficient,not the candidate polynomial's degree;no general existence-uniqueness theorem is invoked.
Rule: ODE_scope_guard · Uses: G001, S027
S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 second-order residual with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute a·y″+b·y′+c·y with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute c·y with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute b·y′ with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute a·y″ with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute y″ Horner add degree0 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute y″ Horner multiply before degree0 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′ Horner add degree0 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute y′ Horner add degree1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute y′ Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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 — Recompute y Horner add degree0 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute y Horner add degree1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute y Horner add degree2 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute y Horner multiply before degree2 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.([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, No)
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7d48c001b31902760a2ba00fbc167cb1c075f8776d9e2fa784d75aa0cef81e12candidate record SHA-256: 94a3e2edfe07727ad02948874c5097c85d232d180dad7e8757d81f015729cad3semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8Lean: NOT_RUN · human review: NOT_PERFORMED