Polynomial IVP Candidates: Initial Data and Finite Residual Grids
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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-20261021-r04-initial-value-problem-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. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-10, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-11. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-10, -3, -2], -2, -12, -11, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -3, -2], -2, -12, -11, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
2. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-9, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-9, -2, 0], -1, -7, -7, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, -2, 0], -1, -7, -7, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
3. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-8, -1, 2], 0, -8, -8, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, -1, 2], 0, -8, -8, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
4. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-7, 0, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-7, 0, -1], 1, -8, -8, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 0, -1], 1, -8, -8, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
5. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-6, 1, 1], 2, 0, 1, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 1, 1], 2, 0, 1, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
6. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-5, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-17. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-5, 2, -2], -2, -17, -17, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, 2, -2], -2, -17, -17, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
7. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-4, 3, 0], -1, -7, -7, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, 3, 0], -1, -7, -7, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
8. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-3, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-3. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-3, -3, 2], 0, -3, -3, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -3, 2], 0, -3, -3, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
9. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-2, -2, -1], 1, -5, -4, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, -2, -1], 1, -5, -4, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
10. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([-1, -1, 1], 2, 1, 1, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, -1, 1], 2, 1, 1, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
11. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[0, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-8. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([0, 0, -2], -2, -8, -8, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0, -2], -2, -8, -8, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
12. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([1, 1, 0], -1, 0, 0, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1, 0], -1, 0, 0, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
13. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[2, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=3. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([2, 2, 2], 0, 2, 3, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2, 2], 0, 2, 3, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
14. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[3, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=5. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([3, 3, -1], 1, 5, 5, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3, -1], 1, 5, 5, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
15. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[4, -3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=2. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([4, -3, 1], 2, 2, 2, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -3, 1], 2, 2, 2, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
16. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[5, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=1. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([5, -2, -2], -2, 1, 1, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, -2, -2], -2, 1, 1, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
17. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[6, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([6, -1, 0], -1, 7, 8, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, -1, 0], -1, 7, 8, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
18. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=7. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([7, 0, 2], 0, 7, 7, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 0, 2], 0, 7, 7, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
19. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([8, 1, -1], 1, 8, 8, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 1, -1], 1, 8, 8, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
20. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[9, 2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=17. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([9, 2, 1], 2, 17, 17, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, 2, 1], 2, 17, 17, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
21. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[10, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-3. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([10, 3, -2], -2, -4, -3, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, 3, -2], -2, -4, -3, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
22. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[11, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=14. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([11, -3, 0], -1, 14, 14, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -3, 0], -1, 14, 14, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
23. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[12, -2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=12. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([12, -2, 2], 0, 12, 12, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, -2, 2], 0, 12, 12, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
24. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[13, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=11. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([13, -1, -1], 1, 11, 11, Yes)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([13, -1, -1], 1, 11, 11, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
25. An Initial Value Check Is Not an ODE Check
Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19. Give (coefficients,x0,y(x0),required y0,match?).
View answer and explanation
([14, 0, 1], 2, 18, 19, No)
Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([14, 0, 1], 2, 18, 19, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-initial-audit · candidate
26. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-10, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-11, required y_prime0=4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-10, -3, -2], -2, -12, 5, -11, 4, No, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -3, -2], -2, -12, 5, -11, 4, No, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
27. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-9, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-9, -2, 0], -1, -7, -2, -7, -2, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, -2, 0], -1, -7, -2, -7, -2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
28. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-8, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-8, -1, 2], 0, -8, -1, -8, -1, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, -1, 2], 0, -8, -1, -8, -1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
29. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-7, 0, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-8, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-7, 0, -1], 1, -8, -2, -8, -3, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 0, -1], 1, -8, -2, -8, -3, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
30. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1, required y_prime0=5. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-6, 1, 1], 2, 0, 5, 1, 5, No, Yes, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 1, 1], 2, 0, 5, 1, 5, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
31. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-5, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-17, required y_prime0=10. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-5, 2, -2], -2, -17, 10, -17, 10, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, 2, -2], -2, -17, 10, -17, 10, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
32. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=-7, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-4, 3, 0], -1, -7, 3, -7, 2, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, 3, 0], -1, -7, 3, -7, 2, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
33. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-3, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=-3, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-3, -3, 2], 0, -3, -3, -3, -3, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -3, 2], 0, -3, -3, -3, -3, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
34. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-2, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=-4, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-2, -2, -1], 1, -5, -4, -4, -4, No, Yes, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, -2, -1], 1, -5, -4, -4, -4, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
35. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[-1, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=1, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([-1, -1, 1], 2, 1, 3, 1, 2, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, -1, 1], 2, 1, 3, 1, 2, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
36. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[0, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-8, required y_prime0=8. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([0, 0, -2], -2, -8, 8, -8, 8, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 0, -2], -2, -8, 8, -8, 8, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
37. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[1, 1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=0, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([1, 1, 0], -1, 0, 1, 0, 1, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 1, 0], -1, 0, 1, 0, 1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
38. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[2, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=3, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([2, 2, 2], 0, 2, 2, 3, 1, No, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, 2, 2], 0, 2, 2, 3, 1, No, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
39. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[3, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=5, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([3, 3, -1], 1, 5, 1, 5, 1, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, 3, -1], 1, 5, 1, 5, 1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
40. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[4, -3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=2, required y_prime0=1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([4, -3, 1], 2, 2, 1, 2, 1, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -3, 1], 2, 2, 1, 2, 1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
41. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[5, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=1, required y_prime0=5. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([5, -2, -2], -2, 1, 6, 1, 5, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, -2, -2], -2, 1, 6, 1, 5, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
42. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[6, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=8, required y_prime0=-1. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([6, -1, 0], -1, 7, -1, 8, -1, No, Yes, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, -1, 0], -1, 7, -1, 8, -1, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
43. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=7, required y_prime0=0. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([7, 0, 2], 0, 7, 0, 7, 0, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 0, 2], 0, 7, 0, 7, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
44. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([8, 1, -1], 1, 8, -1, 8, -2, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 1, -1], 1, 8, -1, 8, -2, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
45. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[9, 2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=17, required y_prime0=6. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([9, 2, 1], 2, 17, 6, 17, 6, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, 2, 1], 2, 17, 6, 17, 6, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
46. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[10, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-2, required y0=-3, required y_prime0=11. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([10, 3, -2], -2, -4, 11, -3, 11, No, Yes, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, 3, -2], -2, -4, 11, -3, 11, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
47. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[11, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=-1, required y0=14, required y_prime0=-4. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([11, -3, 0], -1, 14, -3, 14, -4, Yes, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -3, 0], -1, 14, -3, 14, -4, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
48. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[12, -2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=0, required y0=12, required y_prime0=-2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([12, -2, 2], 0, 12, -2, 12, -2, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, -2, 2], 0, 12, -2, 12, -2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
49. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[13, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=11, required y_prime0=-3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([13, -1, -1], 1, 11, -3, 11, -3, Yes, Yes, Yes)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([13, -1, -1], 1, 11, -3, 11, -3, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
50. Value and Slope Initial Data as Separate Conditions
Use the integer polynomial candidate with ascending-power coefficients=[14, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=2, required y0=19, required y_prime0=3. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
View answer and explanation
([14, 0, 1], 2, 18, 4, 19, 3, No, No, No)
Differentiate exactly and compare both initial quantities separately; one matching condition does not replace the other or verify an ODE. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([14, 0, 1], 2, 18, 4, 19, 3, No, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-initial-audit · candidate
51. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-12, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-2, required y0=-5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, No], [0, -12, -3, -1, No], [1, -15, -3, -1, No]], -2, -6, -5, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-12, -3], 0, -2, [-1, 0, 1], [[-1, -9, -3, -1, No], [0, -12, -3, -1, No], [1, -15, -3, -1, No]], -2, -6, -5, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
52. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-11, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-1, required y0=-9. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, Yes], [0, -11, -2, 0, Yes], [1, -13, -2, 0, Yes]], -1, -9, -9, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-11, -2], 0, -2, [-1, 0, 1], [[-1, -9, -2, 0, Yes], [0, -11, -2, 0, Yes], [1, -13, -2, 0, Yes]], -1, -9, -9, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
53. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-10, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=0, required y0=-10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, Yes], [0, -10, -1, 0, Yes], [1, -11, -1, 0, Yes]], 0, -10, -10, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -1], 0, -1, [-1, 0, 1], [[-1, -9, -1, 0, Yes], [0, -10, -1, 0, Yes], [1, -11, -1, 0, Yes]], 0, -10, -10, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
54. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-9, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=1, required y0=-8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, Yes], [0, -9, 0, 0, Yes], [1, -9, 0, 0, Yes]], 1, -9, -8, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, 0], 0, 0, [-1, 0, 1], [[-1, -9, 0, 0, Yes], [0, -9, 0, 0, Yes], [1, -9, 0, 0, Yes]], 1, -9, -8, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
55. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-8, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=2, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, No], [0, -8, 1, -1, No], [1, -7, 1, -1, No]], 2, -6, -6, Yes, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, 1], 0, 2, [-1, 0, 1], [[-1, -9, 1, -1, No], [0, -8, 1, -1, No], [1, -7, 1, -1, No]], 2, -6, -6, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
56. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-7, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=-2, required y0=-11. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, Yes], [0, -7, 2, 0, Yes], [1, -5, 2, 0, Yes]], -2, -11, -11, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 2], 0, 2, [-1, 0, 1], [[-1, -9, 2, 0, Yes], [0, -7, 2, 0, Yes], [1, -5, 2, 0, Yes]], -2, -11, -11, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
57. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-6, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=-1, required y0=-8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, Yes], [0, -6, 3, 0, Yes], [1, -3, 3, 0, Yes]], -1, -9, -8, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 3], 0, 3, [-1, 0, 1], [[-1, -9, 3, 0, Yes], [0, -6, 3, 0, Yes], [1, -3, 3, 0, Yes]], -1, -9, -8, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
58. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-5, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=0, required y0=-5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, Yes], [0, -5, -3, 0, Yes], [1, -8, -3, 0, Yes]], 0, -5, -5, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, -3], 0, -3, [-1, 0, 1], [[-1, -2, -3, 0, Yes], [0, -5, -3, 0, Yes], [1, -8, -3, 0, Yes]], 0, -5, -5, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
59. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-4, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=-6. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, No], [0, -4, -2, -1, No], [1, -6, -2, -1, No]], 1, -6, -6, Yes, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, -2], 0, -1, [-1, 0, 1], [[-1, -2, -2, -1, No], [0, -4, -2, -1, No], [1, -6, -2, -1, No]], 1, -6, -6, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
60. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=2, required y0=-4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, Yes], [0, -3, -1, 0, Yes], [1, -4, -1, 0, Yes]], 2, -5, -4, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -1], 0, -1, [-1, 0, 1], [[-1, -2, -1, 0, Yes], [0, -3, -1, 0, Yes], [1, -4, -1, 0, Yes]], 2, -5, -4, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
61. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=-2, required y0=-2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, Yes], [0, -2, 0, 0, Yes], [1, -2, 0, 0, Yes]], -2, -2, -2, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 0], 0, 0, [-1, 0, 1], [[-1, -2, 0, 0, Yes], [0, -2, 0, 0, Yes], [1, -2, 0, 0, Yes]], -2, -2, -2, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
62. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[-1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=-1, required y0=-2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, Yes], [0, -1, 1, 0, Yes], [1, 0, 1, 0, Yes]], -1, -2, -2, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 1], 0, 1, [-1, 0, 1], [[-1, -2, 1, 0, Yes], [0, -1, 1, 0, Yes], [1, 0, 1, 0, Yes]], -1, -2, -2, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
63. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=0, required y0=1. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, No], [0, 0, 2, -1, No], [1, 2, 2, -1, No]], 0, 0, 1, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2], 0, 3, [-1, 0, 1], [[-1, -2, 2, -1, No], [0, 0, 2, -1, No], [1, 2, 2, -1, No]], 0, 0, 1, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
64. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[1, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=3; input grid=[-1, 0, 1], x0=1, required y0=4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, Yes], [0, 1, 3, 0, Yes], [1, 4, 3, 0, Yes]], 1, 4, 4, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 3], 0, 3, [-1, 0, 1], [[-1, -2, 3, 0, Yes], [0, 1, 3, 0, Yes], [1, 4, 3, 0, Yes]], 1, 4, 4, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
65. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[2, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=2, required y0=-4. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, Yes], [0, 2, -3, 0, Yes], [1, -1, -3, 0, Yes]], 2, -4, -4, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, -3], 0, -3, [-1, 0, 1], [[-1, 5, -3, 0, Yes], [0, 2, -3, 0, Yes], [1, -1, -3, 0, Yes]], 2, -4, -4, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
66. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=-2, required y0=8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, Yes], [0, 3, -2, 0, Yes], [1, 1, -2, 0, Yes]], -2, 7, 8, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, -2], 0, -2, [-1, 0, 1], [[-1, 5, -2, 0, Yes], [0, 3, -2, 0, Yes], [1, 1, -2, 0, Yes]], -2, 7, 8, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
67. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[4, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=-1, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, No], [0, 4, -1, -1, No], [1, 3, -1, -1, No]], -1, 5, 5, Yes, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -1], 0, 0, [-1, 0, 1], [[-1, 5, -1, -1, No], [0, 4, -1, -1, No], [1, 3, -1, -1, No]], -1, 5, 5, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
68. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[5, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=0, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, Yes], [0, 5, 0, 0, Yes], [1, 5, 0, 0, Yes]], 0, 5, 5, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, Yes], [0, 5, 0, 0, Yes], [1, 5, 0, 0, Yes]], 0, 5, 5, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
69. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[6, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=1, required y0=8. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, Yes], [0, 6, 1, 0, Yes], [1, 7, 1, 0, Yes]], 1, 7, 8, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, 1], 0, 1, [-1, 0, 1], [[-1, 5, 1, 0, Yes], [0, 6, 1, 0, Yes], [1, 7, 1, 0, Yes]], 1, 7, 8, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
70. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[7, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=2; input grid=[-1, 0, 1], x0=2, required y0=11. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, Yes], [0, 7, 2, 0, Yes], [1, 9, 2, 0, Yes]], 2, 11, 11, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 2], 0, 2, [-1, 0, 1], [[-1, 5, 2, 0, Yes], [0, 7, 2, 0, Yes], [1, 9, 2, 0, Yes]], 2, 11, 11, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
71. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[8, 3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=4; input grid=[-1, 0, 1], x0=-2, required y0=2. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, No], [0, 8, 3, -1, No], [1, 11, 3, -1, No]], -2, 2, 2, Yes, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 3], 0, 4, [-1, 0, 1], [[-1, 5, 3, -1, No], [0, 8, 3, -1, No], [1, 11, 3, -1, No]], -2, 2, 2, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
72. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[9, -3]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-3; input grid=[-1, 0, 1], x0=-1, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, Yes], [0, 9, -3, 0, Yes], [1, 6, -3, 0, Yes]], -1, 12, 13, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, -3], 0, -3, [-1, 0, 1], [[-1, 12, -3, 0, Yes], [0, 9, -3, 0, Yes], [1, 6, -3, 0, Yes]], -1, 12, 13, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
73. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[10, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-2; input grid=[-1, 0, 1], x0=0, required y0=10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, Yes], [0, 10, -2, 0, Yes], [1, 8, -2, 0, Yes]], 0, 10, 10, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, -2], 0, -2, [-1, 0, 1], [[-1, 12, -2, 0, Yes], [0, 10, -2, 0, Yes], [1, 8, -2, 0, Yes]], 0, 10, 10, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
74. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[11, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=-1; input grid=[-1, 0, 1], x0=1, required y0=10. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, Yes], [0, 11, -1, 0, Yes], [1, 10, -1, 0, Yes]], 1, 10, 10, Yes, Yes)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -1], 0, -1, [-1, 0, 1], [[-1, 12, -1, 0, Yes], [0, 11, -1, 0, Yes], [1, 10, -1, 0, Yes]], 1, 10, 10, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
75. A First-Order Candidate: Grid Residuals and Initial Value
Use the integer polynomial candidate with ascending-power coefficients=[12, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation y'=p*y+q with p=0, q=1; input grid=[-1, 0, 1], x0=2, required y0=13. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
View answer and explanation
([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, No], [0, 12, 0, -1, No], [1, 12, 0, -1, No]], 2, 12, 13, No, No)
Recompute every grid residual and the initial value independently. Even a final Yes is not a proof on an entire interval. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, 0], 0, 1, [-1, 0, 1], [[-1, 12, 0, -1, No], [0, 12, 0, -1, No], [1, 12, 0, -1, No]], 2, 12, 13, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-first-order-candidate-grid · candidate
76. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-12, -3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-3; input grid=[-1, 0, 1], x0=-2, required y0=-13, required y_prime0=4. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, No], [0, -12, -3, -4, -1, No], [1, -17, -7, -4, -1, No]], -2, -14, 5, -13, 4, No, No, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-12, -3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, -11, 1, -4, -1, No], [0, -12, -3, -4, -1, No], [1, -17, -7, -4, -1, No]], -2, -14, 5, -13, 4, No, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
77. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-11, -2, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-10, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, Yes], [0, -11, -2, -2, 0, Yes], [1, -14, -4, -2, 0, Yes]], -1, -10, 0, -10, 0, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-11, -2, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 0, -2, 0, Yes], [0, -11, -2, -2, 0, Yes], [1, -14, -4, -2, 0, Yes]], -1, -10, 0, -10, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
78. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-10, -1, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-10, required y_prime0=-1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, Yes], [0, -10, -1, 0, 0, Yes], [1, -11, -1, 0, 0, Yes]], 0, -10, -1, -10, -1, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-10, -1, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -9, -1, 0, 0, Yes], [0, -10, -1, 0, 0, Yes], [1, -11, -1, 0, 0, Yes]], 0, -10, -1, -10, -1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
79. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-9, 0, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=-7, required y_prime0=2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, Yes], [0, -9, 0, 2, 0, Yes], [1, -8, 2, 2, 0, Yes]], 1, -8, 2, -7, 2, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-9, 0, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -8, -2, 2, 0, Yes], [0, -9, 0, 2, 0, Yes], [1, -8, 2, 2, 0, Yes]], 1, -8, 2, -7, 2, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
80. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-8, 1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=5; input grid=[-1, 0, 1], x0=2, required y0=2, required y_prime0=9. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, No], [0, -8, 1, 4, -1, No], [1, -5, 5, 4, -1, No]], 2, 2, 9, 2, 9, Yes, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-8, 1, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, -7, -3, 4, -1, No], [0, -8, 1, 4, -1, No], [1, -5, 5, 4, -1, No]], 2, 2, 9, 2, 9, Yes, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
81. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-7, 2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=-19, required y_prime0=9. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, Yes], [0, -7, 2, -4, 0, Yes], [1, -7, -2, -4, 0, Yes]], -2, -19, 10, -19, 9, Yes, No, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-7, 2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -11, 6, -4, 0, Yes], [0, -7, 2, -4, 0, Yes], [1, -7, -2, -4, 0, Yes]], -2, -19, 10, -19, 9, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
82. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-6, 3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-9, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, Yes], [0, -6, 3, -2, 0, Yes], [1, -4, 1, -2, 0, Yes]], -1, -10, 5, -9, 5, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-6, 3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -10, 5, -2, 0, Yes], [0, -6, 3, -2, 0, Yes], [1, -4, 1, -2, 0, Yes]], -1, -10, 5, -9, 5, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
83. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-5, -3, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=-5, required y_prime0=-3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, Yes], [0, -5, -3, 0, 0, Yes], [1, -8, -3, 0, 0, Yes]], 0, -5, -3, -5, -3, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-5, -3, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, -2, -3, 0, 0, Yes], [0, -5, -3, 0, 0, Yes], [1, -8, -3, 0, 0, Yes]], 0, -5, -3, -5, -3, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
84. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-4, -2, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=3; input grid=[-1, 0, 1], x0=1, required y0=-5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-4, -2, 1], 1, 0, 0, 3, [-1, 0, 1], [[-1, -1, -4, 2, -1, No], [0, -4, -2, 2, -1, No], [1, -5, 0, 2, -1, No]], 1, -5, 0, -5, 0, Yes, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
85. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-3, -1, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=4, required y_prime0=7. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, Yes], [0, -3, -1, 4, 0, Yes], [1, -2, 3, 4, 0, Yes]], 2, 3, 7, 4, 7, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-3, -1, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 0, -5, 4, 0, Yes], [0, -3, -1, 4, 0, Yes], [1, -2, 3, 4, 0, Yes]], 2, 3, 7, 4, 7, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
86. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-2, 0, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=-10, required y_prime0=7. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, Yes], [0, -2, 0, -4, 0, Yes], [1, -4, -4, -4, 0, Yes]], -2, -10, 8, -10, 7, Yes, No, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-2, 0, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, -4, 4, -4, 0, Yes], [0, -2, 0, -4, 0, Yes], [1, -4, -4, -4, 0, Yes]], -2, -10, 8, -10, 7, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
87. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[-1, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=-3, required y_prime0=3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, Yes], [0, -1, 1, -2, 0, Yes], [1, -1, -1, -2, 0, Yes]], -1, -3, 3, -3, 3, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([-1, 1, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, -3, 3, -2, 0, Yes], [0, -1, 1, -2, 0, Yes], [1, -1, -1, -2, 0, Yes]], -1, -3, 3, -3, 3, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
88. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[0, 2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=1; input grid=[-1, 0, 1], x0=0, required y0=1, required y_prime0=2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, No], [0, 0, 2, 0, -1, No], [1, 2, 2, 0, -1, No]], 0, 0, 2, 1, 2, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([0, 2, 0], 1, 0, 0, 1, [-1, 0, 1], [[-1, -2, 2, 0, -1, No], [0, 0, 2, 0, -1, No], [1, 2, 2, 0, -1, No]], 0, 0, 2, 1, 2, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
89. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[1, 3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=5, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, Yes], [0, 1, 3, 2, 0, Yes], [1, 5, 5, 2, 0, Yes]], 1, 5, 5, 5, 5, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, Yes], [0, 1, 3, 2, 0, Yes], [1, 5, 5, 2, 0, Yes]], 1, 5, 5, 5, 5, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
90. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[2, -3, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=4, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, Yes], [0, 2, -3, 4, 0, Yes], [1, 1, 1, 4, 0, Yes]], 2, 4, 5, 4, 5, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([2, -3, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -7, 4, 0, Yes], [0, 2, -3, 4, 0, Yes], [1, 1, 1, 4, 0, Yes]], 2, 4, 5, 4, 5, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
91. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[3, -2, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-4; input grid=[-1, 0, 1], x0=-2, required y0=0, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, Yes], [0, 3, -2, -4, 0, Yes], [1, -1, -6, -4, 0, Yes]], -2, -1, 6, 0, 5, No, No, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([3, -2, -2], 1, 0, 0, -4, [-1, 0, 1], [[-1, 3, 2, -4, 0, Yes], [0, 3, -2, -4, 0, Yes], [1, -1, -6, -4, 0, Yes]], -2, -1, 6, 0, 5, No, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
92. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[4, -1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-1; input grid=[-1, 0, 1], x0=-1, required y0=4, required y_prime0=1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, No], [0, 4, -1, -2, -1, No], [1, 2, -3, -2, -1, No]], -1, 4, 1, 4, 1, Yes, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([4, -1, -1], 1, 0, 0, -1, [-1, 0, 1], [[-1, 4, 1, -2, -1, No], [0, 4, -1, -2, -1, No], [1, 2, -3, -2, -1, No]], -1, 4, 1, 4, 1, Yes, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
93. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[5, 0, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=5, required y_prime0=0. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, Yes], [0, 5, 0, 0, 0, Yes], [1, 5, 0, 0, 0, Yes]], 0, 5, 0, 5, 0, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([5, 0, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, 0, Yes], [0, 5, 0, 0, 0, Yes], [1, 5, 0, 0, 0, Yes]], 0, 5, 0, 5, 0, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
94. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[6, 1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=9, required y_prime0=3. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, Yes], [0, 6, 1, 2, 0, Yes], [1, 8, 3, 2, 0, Yes]], 1, 8, 3, 9, 3, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([6, 1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 6, -1, 2, 0, Yes], [0, 6, 1, 2, 0, Yes], [1, 8, 3, 2, 0, Yes]], 1, 8, 3, 9, 3, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
95. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[7, 2, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=4; input grid=[-1, 0, 1], x0=2, required y0=19, required y_prime0=10. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, Yes], [0, 7, 2, 4, 0, Yes], [1, 11, 6, 4, 0, Yes]], 2, 19, 10, 19, 10, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([7, 2, 2], 1, 0, 0, 4, [-1, 0, 1], [[-1, 7, -2, 4, 0, Yes], [0, 7, 2, 4, 0, Yes], [1, 11, 6, 4, 0, Yes]], 2, 19, 10, 19, 10, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
96. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[8, 3, -2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-3; input grid=[-1, 0, 1], x0=-2, required y0=-6, required y_prime0=10. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, No], [0, 8, 3, -4, -1, No], [1, 9, -1, -4, -1, No]], -2, -6, 11, -6, 10, Yes, No, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 3, -2], 1, 0, 0, -3, [-1, 0, 1], [[-1, 3, 7, -4, -1, No], [0, 8, 3, -4, -1, No], [1, 9, -1, -4, -1, No]], -2, -6, 11, -6, 10, Yes, No, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
97. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[9, -3, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=-2; input grid=[-1, 0, 1], x0=-1, required y0=12, required y_prime0=-1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, Yes], [0, 9, -3, -2, 0, Yes], [1, 5, -5, -2, 0, Yes]], -1, 11, -1, 12, -1, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([9, -3, -1], 1, 0, 0, -2, [-1, 0, 1], [[-1, 11, -1, -2, 0, Yes], [0, 9, -3, -2, 0, Yes], [1, 5, -5, -2, 0, Yes]], -1, 11, -1, 12, -1, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
98. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[10, -2, 0]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=0; input grid=[-1, 0, 1], x0=0, required y0=10, required y_prime0=-2. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, Yes], [0, 10, -2, 0, 0, Yes], [1, 8, -2, 0, 0, Yes]], 0, 10, -2, 10, -2, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([10, -2, 0], 1, 0, 0, 0, [-1, 0, 1], [[-1, 12, -2, 0, 0, Yes], [0, 10, -2, 0, 0, Yes], [1, 8, -2, 0, 0, Yes]], 0, 10, -2, 10, -2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
99. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[11, -1, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=11, required y_prime0=1. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, Yes], [0, 11, -1, 2, 0, Yes], [1, 11, 1, 2, 0, Yes]], 1, 11, 1, 11, 1, Yes, Yes, Yes)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([11, -1, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, 13, -3, 2, 0, Yes], [0, 11, -1, 2, 0, Yes], [1, 11, 1, 2, 0, Yes]], 1, 11, 1, 11, 1, Yes, Yes, Yes).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
100. A Second-Order Candidate with Two Initial Data
Use the integer polynomial candidate with ascending-power coefficients=[12, 0, 2]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=5; input grid=[-1, 0, 1], x0=2, required y0=21, required y_prime0=8. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).
View answer and explanation
([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, No], [0, 12, 0, 4, -1, No], [1, 14, 4, 4, -1, No]], 2, 20, 8, 21, 8, No, Yes, No)
Recompute both derivatives, residual rows and initial values. Finite agreement is not an identity on an entire interval or an existence/uniqueness theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([12, 0, 2], 1, 0, 0, 5, [-1, 0, 1], [[-1, 14, -4, 4, -1, No], [0, 12, 0, 4, -1, No], [1, 14, 4, 4, -1, No]], 2, 20, 8, 21, 8, No, Yes, No).
Reviewed step-by-step derivation
Initial-value problem · qf-app21-ivp-second-order-candidate-grid · candidate
Canonical question JSONL · English JSONL · Validation report