Polynomial ODE Candidates: Point Residuals and Initial Conditions
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Ordinary differential equation · qf-app20-ode-initial-condition-audit · candidate
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