# Polynomial ODE Candidates: Point Residuals and Initial Conditions

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0], 0, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0], 0, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0001/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 2. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1], 1, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1], 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0002/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 3. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1], 1, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1], 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0003/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 4. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([2], 2, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2], 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0004/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 5. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 2], 2, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2], 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0005/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 6. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1, 2], 2, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2], 2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0006/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 7. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1, 2], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1, 2], 2, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2], 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0007/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 8. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([3], 3, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3], 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0008/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 9. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 3], 3, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3], 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0009/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 10. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1, 3], 3, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 3], 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0010/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 11. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1, 3], 3, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0011/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 12. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([2, 3], 3, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 3], 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0012/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 13. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 2, 3], 3, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0013/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 14. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1, 2, 3], 3, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 3], 3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0014/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 15. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1, 2, 3], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1, 2, 3], 3, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2, 3], 3, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0015/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 16. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([4], 4, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4], 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0016/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 17. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 4], 4, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 4], 4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0017/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 18. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1, 4], 4, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 4], 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0018/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 19. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1, 4], 4, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 4], 4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0019/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 20. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([2, 4], 4, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 4], 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0020/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 21. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 2, 4], 4, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 4], 4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0021/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 22. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [1, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([1, 2, 4], 4, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 4], 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0022/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 23. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 1, 2, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 1, 2, 4], 4, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 2, 4], 4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0023/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 24. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [3, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([3, 4], 4, No)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 4], 4, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0024/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 25. Highest Explicit Derivative Order, Not Candidate Polynomial Degree

Derivative orders occurring in the specified relation are [0, 3, 4], with 0 denoting y itself. Give (sorted distinct orders, highest order, does y occur?). A [0]-only list records the algebraic order-zero boundary, not evidence of a positive-order ODE.

View answer and explanation([0, 3, 4], 4, Yes)

Use the derivatives actually declared in the relation, not the polynomial degree or number of terms. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3, 4], 4, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0025/)

Ordinary differential equation · qf-app20-ode-order-classification · candidate





## 26. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-12, 1, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=-20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-12, 1, -3], -3, -42, 19, -3, -20, -87, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-12, 1, -3], -3, -42, 19, -3, -20, -87, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0026/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 27. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-11, 2, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=-18. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-11, 2, -1], -2, -19, 6, -2, -18, -14, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-11, 2, -1], -2, -19, 6, -2, -18, -14, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0027/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 28. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-10, 3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=-16. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-10, 3, 1], -1, -12, 1, -1, -16, 5, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-10, 3, 1], -1, -12, 1, -1, -16, 5, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0028/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 29. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-9, 4, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=-14. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-9, 4, 3], 0, -9, 4, 0, -14, 18, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, 4, 3], 0, -9, 4, 0, -14, 18, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0029/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 30. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, 5, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=-12. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-8, 5, -2], 1, -5, 1, 1, -12, 18, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, 5, -2], 1, -5, 1, 1, -12, 18, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0030/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 31. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=-10. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-7, 1, 0], 2, -5, 1, 2, -10, 21, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, 1, 0], 2, -5, 1, 2, -10, 21, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0031/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 32. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=-8. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-6, 2, 2], 3, 18, 14, 3, -8, -32, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 2, 2], 3, 18, 14, 3, -8, -32, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0032/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 33. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 3, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=-6. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-5, 3, -3], -3, -41, 21, -3, -6, -96, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 3, -3], -3, -41, 21, -3, -6, -96, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0033/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 34. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 4, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=-4. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-4, 4, -1], -2, -16, 8, -2, -4, -20, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 4, -1], -2, -16, 8, -2, -4, -20, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0034/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 35. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, 5, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=-2. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-3, 5, 1], -1, -7, 3, -1, -2, -2, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 5, 1], -1, -7, 3, -1, -2, -2, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0035/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 36. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, 1, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=0. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-2, 1, 3], 0, -2, 1, 0, 0, 1, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, 3], 0, -2, 1, 0, 0, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0036/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 37. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, 2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=2. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([-1, 2, -2], 1, -1, -2, 1, 2, -3, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, -2], 1, -1, -2, 1, 2, -3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0037/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 38. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[0, 3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=4. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([0, 3, 0], 2, 6, 3, 2, 4, -13, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 3, 0], 2, 6, 3, 2, 4, -13, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0038/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 39. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 4, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=6. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([1, 4, 2], 3, 31, 16, 3, 6, -83, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 4, 2], 3, 31, 16, 3, 6, -83, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0039/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 40. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[2, 5, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=8. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([2, 5, -3], -3, -40, 23, -3, 8, -105, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 5, -3], -3, -40, 23, -3, 8, -105, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0040/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 41. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[3, 1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=10. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([3, 1, -1], -2, -3, 5, -2, 10, -11, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, -1], -2, -3, 5, -2, 10, -11, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0041/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 42. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=12. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([4, 2, 1], -1, 3, 0, -1, 12, -9, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 2, 1], -1, 3, 0, -1, 12, -9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0042/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 43. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[5, 3, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=14. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([5, 3, 3], 0, 5, 3, 0, 14, -11, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 3, 3], 0, 5, 3, 0, 14, -11, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0043/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 44. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[6, 4, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=1, check y'=p*y+q with p=1, q=16. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([6, 4, -2], 1, 8, 0, 1, 16, -24, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 4, -2], 1, 8, 0, 1, 16, -24, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0044/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 45. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[7, 5, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=2, check y'=p*y+q with p=2, q=18. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([7, 5, 0], 2, 17, 5, 2, 18, -47, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, 5, 0], 2, 17, 5, 2, 18, -47, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0045/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 46. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([8, 1, 2], 3, 29, 13, 3, 20, -94, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, 1, 2], 3, 29, 13, 3, 20, -94, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0046/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 47. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 2, -3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-3, check y'=p*y+q with p=-3, q=22. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([9, 2, -3], -3, -24, 20, -3, 22, -74, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 2, -3], -3, -24, 20, -3, 22, -74, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0047/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 48. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 3, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-2, check y'=p*y+q with p=-2, q=24. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([10, 3, -1], -2, 0, 7, -2, 24, -17, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 3, -1], -2, 0, 7, -2, 24, -17, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0048/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 49. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 4, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=-1, check y'=p*y+q with p=-1, q=26. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([11, 4, 1], -1, 8, 2, -1, 26, -16, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 4, 1], -1, 8, 2, -1, 26, -16, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0049/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 50. One-Point Residual for a First-Order Relation

The integer-coefficient polynomial candidate has ascending-power coefficients=[12, 5, 3]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=0, check y'=p*y+q with p=0, q=28. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).

View answer and explanation([12, 5, 3], 0, 12, 5, 0, 28, -23, No)

Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, 5, 3], 0, 12, 5, 0, 28, -23, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0050/)

Ordinary differential equation · qf-app20-ode-first-order-residual · candidate





## 51. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, -2, -3, 0]. 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=-4, rhs=-20. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, -3, 0], -2, -16, 10, -6, 1, -3, -4, -20, 48, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0051/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 52. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, -1, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-2, c=-3, rhs=-17. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, -1, 0, 1], -1, -7, 2, -6, 2, -2, -3, -17, 22, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0052/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 53. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 0, 3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-1, c=-2, rhs=-14. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 0, 3, 2], 0, -6, 0, 6, 3, -1, -2, -14, 44, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0053/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 54. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 1, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=0, c=-1, rhs=-11. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 1, -1, 0], 1, -5, -1, -2, 4, 0, -1, -11, 8, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0054/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 55. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 2, 2, 1]. 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=5≠0, b=1, c=0, rhs=-8. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 2, 2, 1], 2, 16, 22, 16, 5, 1, 0, -8, 110, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0055/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 56. One-Point Residual with a Nonzero Second-Order Coefficient

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?).

View answer and explanation([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -2, -2, 2], -2, -23, 30, -28, 1, 2, 1, -5, 14, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0056/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 57. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, -1, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=3, c=2, rhs=-2. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, 1, 0], -1, 0, -3, 2, 2, 3, 2, -2, -3, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0057/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 58. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, 0, -3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-3, c=3, rhs=1. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, -3, 1], 0, -1, 0, -6, 3, -3, 3, 1, -22, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0058/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 59. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[0, 1, 0, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=-2, c=4, rhs=4. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 0, 2], 1, 3, 7, 12, 4, -2, 4, 4, 42, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0059/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 60. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 2, 3, 0]. 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=5≠0, b=-1, c=-4, rhs=7. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 2, 3, 0], 2, 17, 14, 6, 5, -1, -4, 7, -59, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0060/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 61. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[2, -2, -1, 1]. 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=0, c=-3, rhs=10. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -2, -1, 1], -2, -6, 14, -14, 1, 0, -3, 10, -6, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0061/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 62. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[3, -1, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=1, c=-2, rhs=13. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2, 2], -1, 4, 1, -8, 2, 1, -2, 13, -36, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0062/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 63. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 0, -2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=2, c=-1, rhs=16. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 0, -2, 0], 0, 4, 0, -4, 3, 2, -1, 16, -32, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0063/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 64. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[5, 1, 1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=3, c=0, rhs=19. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, 1, 1, 1], 1, 8, 6, 8, 4, 3, 0, 19, 31, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0064/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 65. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[6, 2, -3, 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=5≠0, b=-3, c=1, rhs=22. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, 2, -3, 2], 2, 14, 14, 18, 5, -3, 1, 22, 40, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0065/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 66. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[7, -2, 0, 0]. 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=2, rhs=25. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -2, 0, 0], -2, 11, -2, 0, 1, -2, 2, 25, 1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0066/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 67. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[8, -1, 3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-1, c=3, rhs=28. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, -1, 3, 1], -1, 11, -4, 0, 2, -1, 3, 28, 9, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0067/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 68. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 0, -1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=0, c=4, rhs=31. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 0, -1, 2], 0, 9, 0, -2, 3, 0, 4, 31, -1, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0068/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 69. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 1, 2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=1, c=-4, rhs=34. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 1, 2, 0], 1, 13, 5, 4, 4, 1, -4, 34, -65, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0069/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 70. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 2, -2, 1]. 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=5≠0, b=2, c=-3, rhs=37. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 2, -2, 1], 2, 15, 6, 8, 5, 2, -3, 37, -30, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0070/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 71. One-Point Residual with a Nonzero Second-Order Coefficient

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?).

View answer and explanation([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, -2, 1, 2], -2, 4, 18, -22, 1, 3, -2, 40, -16, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0071/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 72. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[13, -1, -3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=-1, check a*y''+b*y'+c*y=rhs with a=2≠0, b=-3, c=-1, rhs=43. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -1, -3, 0], -1, 11, 5, -6, 2, -3, -1, 43, -81, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0072/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 73. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[14, 0, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=0, check a*y''+b*y'+c*y=rhs with a=3≠0, b=-2, c=0, rhs=46. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, 0, 0, 1], 0, 14, 0, 0, 3, -2, 0, 46, -46, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0073/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 74. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[15, 1, 3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At x=1, check a*y''+b*y'+c*y=rhs with a=4≠0, b=-1, c=1, rhs=49. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([15, 1, 3, 2], 1, 21, 13, 18, 4, -1, 1, 49, 31, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0074/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 75. One-Point Residual with a Nonzero Second-Order Coefficient

The integer-coefficient polynomial candidate has ascending-power coefficients=[16, 2, -1, 0]. 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=5≠0, b=0, c=2, rhs=52. Give (coefficients, x, y, y', y'', a, b, c, rhs, signed residual, zero?).

View answer and explanation([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, No)

Recompute both derivatives and the signed equation residual; no existence or uniqueness theorem follows. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([16, 2, -1, 0], 2, 16, -2, -2, 5, 0, 2, 52, -30, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0075/)

Ordinary differential equation · qf-app20-ode-second-order-residual · candidate





## 76. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-9, -3, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-10, required y'0=4. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-9, -3, -2], -2, -11, 5, -10, 4, No, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-9, -3, -2], -2, -11, 5, -10, 4, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0076/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 77. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-8, -2, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=-6, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-8, -2, 0], -1, -6, -2, -6, -2, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-8, -2, 0], -1, -6, -2, -6, -2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0077/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 78. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-7, -1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=-7, required y'0=-1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-7, -1, 2], 0, -7, -1, -7, -1, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-7, -1, 2], 0, -7, -1, -7, -1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0078/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 79. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-6, 0, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=-6, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-6, 0, -1], 1, -7, -2, -6, -2, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-6, 0, -1], 1, -7, -2, -6, -2, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0079/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 80. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-5, 1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=1, required y'0=4. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-5, 1, 1], 2, 1, 5, 1, 4, Yes, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-5, 1, 1], 2, 1, 5, 1, 4, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0080/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 81. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-4, 2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-16, required y'0=10. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-4, 2, -2], -2, -16, 10, -16, 10, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-4, 2, -2], -2, -16, 10, -16, 10, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0081/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 82. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-3, 3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=-5, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-3, 3, 0], -1, -6, 3, -5, 3, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 3, 0], -1, -6, 3, -5, 3, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0082/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 83. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-2, -3, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=-2, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-2, -3, 2], 0, -2, -3, -2, -3, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -3, 2], 0, -2, -3, -2, -3, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0083/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 84. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[-1, -2, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=-4, required y'0=-5. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([-1, -2, -1], 1, -4, -4, -4, -5, Yes, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -2, -1], 1, -4, -4, -4, -5, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0084/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 85. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[0, -1, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=3, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([0, -1, 1], 2, 2, 3, 3, 3, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 1], 2, 2, 3, 3, 3, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0085/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 86. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[1, 0, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-7, required y'0=8. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([1, 0, -2], -2, -7, 8, -7, 8, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], -2, -7, 8, -7, 8, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0086/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 87. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[2, 1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=1, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([2, 1, 0], -1, 1, 1, 1, 1, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], -1, 1, 1, 1, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0087/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 88. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[3, 2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=4, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([3, 2, 2], 0, 3, 2, 4, 1, No, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 2, 2], 0, 3, 2, 4, 1, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0088/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 89. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[4, 3, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=6, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([4, 3, -1], 1, 6, 1, 6, 1, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([4, 3, -1], 1, 6, 1, 6, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0089/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 90. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[5, -3, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=3, required y'0=1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([5, -3, 1], 2, 3, 1, 3, 1, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([5, -3, 1], 2, 3, 1, 3, 1, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0090/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 91. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[6, -2, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=3, required y'0=6. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([6, -2, -2], -2, 2, 6, 3, 6, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([6, -2, -2], -2, 2, 6, 3, 6, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0091/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 92. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[7, -1, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=8, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([7, -1, 0], -1, 8, -1, 8, -2, Yes, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([7, -1, 0], -1, 8, -1, 8, -2, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0092/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 93. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 0, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=8, required y'0=0. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([8, 0, 2], 0, 8, 0, 8, 0, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, 0, 2], 0, 8, 0, 8, 0, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0093/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 94. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[9, 1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=10, required y'0=-1. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([9, 1, -1], 1, 9, -1, 10, -1, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([9, 1, -1], 1, 9, -1, 10, -1, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0094/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 95. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[10, 2, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=18, required y'0=6. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([10, 2, 1], 2, 18, 6, 18, 6, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([10, 2, 1], 2, 18, 6, 18, 6, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0095/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 96. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[11, 3, -2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-2, required y0=-3, required y'0=10. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([11, 3, -2], -2, -3, 11, -3, 10, Yes, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([11, 3, -2], -2, -3, 11, -3, 10, Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0096/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 97. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[12, -3, 0]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=-1, required y0=16, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([12, -3, 0], -1, 15, -3, 16, -3, No, Yes, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([12, -3, 0], -1, 15, -3, 16, -3, No, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0097/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 98. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[13, -2, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=0, required y0=13, required y'0=-2. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([13, -2, 2], 0, 13, -2, 13, -2, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([13, -2, 2], 0, 13, -2, 13, -2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0098/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 99. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[14, -1, -1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=1, required y0=12, required y'0=-3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([14, -1, -1], 1, 12, -3, 12, -3, Yes, Yes, Yes)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([14, -1, -1], 1, 12, -3, 12, -3, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0099/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate





## 100. Candidate Value and Derivative against Two Initial Conditions

The integer-coefficient polynomial candidate has ascending-power coefficients=[15, 0, 1]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. x0=2, required y0=20, required y'0=3. Give (coefficients, x0, actual y, actual y', required y0, required y'0, first condition?, second condition?, both?).

View answer and explanation([15, 0, 1], 2, 19, 4, 20, 3, No, No, No)

Audit both stated initial conditions separately. This test alone does not check an ODE or prove an initial-value problem solvable. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([15, 0, 1], 2, 19, 4, 20, 3, No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261020-r02-ordinary-differential-equation-0100/)

Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations/validation-report.json)