# Question 1981: One-Point Residual with a Nonzero Second-Order Coefficient

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0002-01981 · q-20261013-advance-20261020-r02-ordinary-differential-equation-0056



## Original question and givens

The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, -2, -2, 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=2, c=1, rhs=-5. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).


- G001 — The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, -2, -2, 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=2, c=1, rhs=-5. 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](https://ocme.evemisslab.com/en/mko/mko-ordinary-differential-equation/)





## Step-by-step derivation

S001 · impliesBefore

candidate y ascending coefficients=[-3, -2, -2, 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.

Step identifiers and dependenciesRule: polynomial_candidate_domain · Uses: G001





S002 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

y″ coefficient at degree0 := mul(1, -4)

After

y″ coefficient at degree0 = -4

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

y ascending coefficients=[-3, -2, -2, 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.

Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S006





S008 · calculationBefore

y Horner multiply before degree2 := mul(2, -2)

After

y Horner multiply before degree2 = -4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

y Horner add degree2 := sum(-4, -2)

After

y Horner add degree2 = -6

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

y Horner multiply before degree1 := mul(-6, -2)

After

y Horner multiply before degree1 = 12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

y Horner add degree1 := sum(12, -2)

After

y Horner add degree1 = 10

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

y Horner multiply before degree0 := mul(10, -2)

After

y Horner multiply before degree0 = -20

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

y Horner add degree0 := sum(-20, -3)

After

y Horner add degree0 = -23

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

y′ ascending coefficients=[-2, -4, 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.

Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S013





S015 · calculationBefore

y′ Horner multiply before degree1 := mul(6, -2)

After

y′ Horner multiply before degree1 = -12

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

y′ Horner add degree1 := sum(-12, -4)

After

y′ Horner add degree1 = -16

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

y′ Horner multiply before degree0 := mul(-16, -2)

After

y′ Horner multiply before degree0 = 32

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

y′ Horner add degree0 := sum(32, -2)

After

y′ Horner add degree0 = 30

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · impliesBefore

y″ ascending coefficients=[-4, 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.

Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S018





S020 · calculationBefore

y″ Horner multiply before degree0 := mul(12, -2)

After

y″ Horner multiply before degree0 = -24

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

y″ Horner add degree0 := sum(-24, -4)

After

y″ Horner add degree0 = -28

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

a·y″ := mul(1, -28)

After

a·y″ = -28

Conditions: none

Multiply each derivative by its own original equation coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

b·y′ := mul(2, 30)

After

b·y′ = 60

Conditions: none

Multiply each derivative by its own original equation coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

c·y := mul(1, -23)

After

c·y = -23

Conditions: none

Multiply each derivative by its own original equation coefficient.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

a·y″+b·y′+c·y := sum(-28, 60, -23)

After

a·y″+b·y′+c·y = 9

Conditions: none

Sum all three terms before comparing with rhs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

second-order residual := sub(9, -5)

After

second-order residual = 14

Conditions: none

Compute the residual with the same left-minus-right sign convention.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · impliesBefore

residual at x=-2 is 14

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.

Step identifiers and dependenciesRule: point_residual_verdict · Uses: G001, S026





S028 · impliesBefore

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.

Step identifiers and dependenciesRule: ODE_scope_guard · Uses: G001, S027









## Reverse checks


- 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.





## Result

([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 6717d2705f3bc8fd6c56a1bbecafdee40dc4f8272dc06773b6853b20250b038d
- candidate record SHA-256: 710ce48467988cb5501756fabf70e6177378fd0b0251bb48ca573a71b8007d7b
- semantic review receipt SHA-256: 340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861
- question-first witness SHA-256: 9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8
- Lean: NOT_RUN · human review: NOT_PERFORMED

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