Explicit PDE Conventions: Point Residuals and Symbolic Identities
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batch-20261013-advance-20261024-r06-partial-differential-equation-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-12, -2, -3, 1, -1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, -2] means a*u_x+b*u_y; source=-12, x=-2, y=-3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-12, -2, -3, 1, -1, -2], [1, -2], -2, -3, -3, 11, -12, -13, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [1, -2], -2, -3, -3, 11, -12, -13, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
2. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-11, -1, -2, 2, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, -1] means a*u_x+b*u_y; source=-11, x=-1, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-11, -1, -2, 2, 0, -1], [2, -1], -1, -2, -5, 2, -11, -1, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [2, -1], -1, -2, -5, 2, -11, -1, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
3. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-10, 0, -1, 3, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, 0] means a*u_x+b*u_y; source=-10, x=0, y=-1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-10, 0, -1, 3, 1, 0], [3, 0], 0, -1, -1, -1, -10, 7, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [3, 0], 0, -1, -1, -1, -10, 7, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
4. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-9, 1, 0, 4, -1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, 1] means a*u_x+b*u_y; source=-9, x=1, y=0. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-9, 1, 0, 4, -1, 1], [1, 1], 1, 0, 9, -1, -9, 17, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [1, 1], 1, 0, 9, -1, -9, 17, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
5. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-8, 2, 1, 1, 0, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, 2] means a*u_x+b*u_y; source=-8, x=2, y=1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-8, 2, 1, 1, 0, 2], [2, 2], 2, 1, 6, 5, -8, 30, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [2, 2], 2, 1, 6, 5, -8, 30, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
6. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-7, -2, 2, 2, 1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, -2] means a*u_x+b*u_y; source=-7, x=-2, y=2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-7, -2, 2, 2, 1, -2], [3, -2], -2, 2, -8, -8, -7, -1, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [3, -2], -2, 2, -8, -8, -7, -1, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
7. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-6, -1, 3, 3, -1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, -1] means a*u_x+b*u_y; source=-6, x=-1, y=3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-6, -1, 3, 3, -1, -1], [1, -1], -1, 3, -10, -2, -6, -2, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [1, -1], -1, 3, -10, -2, -6, -2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
8. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-5, 0, -3, 4, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, 0] means a*u_x+b*u_y; source=-5, x=0, y=-3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-5, 0, -3, 4, 0, 0], [2, 0], 0, -3, 0, -3, -5, 5, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [2, 0], 0, -3, 0, -3, -5, 5, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
9. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-4, 1, -2, 1, 1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, 1] means a*u_x+b*u_y; source=-4, x=1, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-4, 1, -2, 1, 1, 1], [3, 1], 1, -2, 1, -5, -4, 2, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [3, 1], 1, -2, 1, -5, -4, 2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
10. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-3, 2, -1, 2, -1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, 2] means a*u_x+b*u_y; source=-3, x=2, y=-1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-3, 2, -1, 2, -1, 2], [1, 2], 2, -1, 11, -7, -3, 0, Yes)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [1, 2], 2, -1, 11, -7, -3, 0, Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
11. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-2, -2, 0, 3, 0, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, -2] means a*u_x+b*u_y; source=-2, x=-2, y=0. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-2, -2, 0, 3, 0, -2], [2, -2], -2, 0, -14, 0, -2, -26, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [2, -2], -2, 0, -14, 0, -2, -26, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
12. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 1, 4, 1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, -1] means a*u_x+b*u_y; source=-1, x=-1, y=1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([-1, -1, 1, 4, 1, -1], [3, -1], -1, 1, -8, -2, -1, -21, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [3, -1], -1, 1, -8, -2, -1, -21, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
13. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, -1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, 0] means a*u_x+b*u_y; source=0, x=0, y=2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([0, 0, 2, 1, -1, 0], [1, 0], 0, 2, -2, 2, 0, -2, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [1, 0], 0, 2, -2, 2, 0, -2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
14. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[1, 1, 3, 2, 0, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, 1] means a*u_x+b*u_y; source=1, x=1, y=3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([1, 1, 3, 2, 0, 1], [2, 1], 1, 3, 5, 9, 1, 18, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [2, 1], 1, 3, 5, 9, 1, 18, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
15. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[2, 2, -3, 3, 1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, 2] means a*u_x+b*u_y; source=2, x=2, y=-3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([2, 2, -3, 3, 1, 2], [3, 2], 2, -3, 11, -13, 2, 5, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [3, 2], 2, -3, 11, -13, 2, 5, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
16. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[3, -2, -2, 4, -1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, -2] means a*u_x+b*u_y; source=3, x=-2, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([3, -2, -2, 4, -1, -2], [1, -2], -2, -2, -16, 8, 3, -35, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [1, -2], -2, -2, -16, 8, 3, -35, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
17. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[4, -1, -1, 1, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, -1] means a*u_x+b*u_y; source=4, x=-1, y=-1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([4, -1, -1, 1, 0, -1], [2, -1], -1, -1, -3, 1, 4, -11, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [2, -1], -1, -1, -3, 1, 4, -11, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
18. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[5, 0, 0, 2, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, 0] means a*u_x+b*u_y; source=5, x=0, y=0. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([5, 0, 0, 2, 1, 0], [3, 0], 0, 0, 0, 0, 5, -5, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [3, 0], 0, 0, 0, 0, 5, -5, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
19. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[6, 1, 1, 3, -1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, 1] means a*u_x+b*u_y; source=6, x=1, y=1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([6, 1, 1, 3, -1, 1], [1, 1], 1, 1, 6, 2, 6, 2, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, 1], 1, 1, 6, 2, 6, 2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
20. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[7, 2, 2, 4, 0, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, 2] means a*u_x+b*u_y; source=7, x=2, y=2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([7, 2, 2, 4, 0, 2], [2, 2], 2, 2, 18, 10, 7, 49, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 2], 2, 2, 18, 10, 7, 49, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
21. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[8, -2, 3, 1, 1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, -2] means a*u_x+b*u_y; source=8, x=-2, y=3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([8, -2, 3, 1, 1, -2], [3, -2], -2, 3, -3, -11, 8, 5, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [3, -2], -2, 3, -3, -11, 8, 5, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
22. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[9, -1, -3, 2, -1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, -1] means a*u_x+b*u_y; source=9, x=-1, y=-3. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([9, -1, -3, 2, -1, -1], [1, -1], -1, -3, -2, 4, 9, -15, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [1, -1], -1, -3, -2, 4, 9, -15, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
23. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[10, 0, -2, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[2, 0] means a*u_x+b*u_y; source=10, x=0, y=-2. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([10, 0, -2, 3, 0, 0], [2, 0], 0, -2, 0, -2, 10, -10, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [2, 0], 0, -2, 0, -2, 10, -10, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
24. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[11, 1, -1, 4, 1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[3, 1] means a*u_x+b*u_y; source=11, x=1, y=-1. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([11, 1, -1, 4, 1, 1], [3, 1], 1, -1, 8, -2, 11, 11, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [3, 1], 1, -1, 8, -2, 11, 11, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
25. One-Point Residual of a Declared First-Order PDE Operator
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[12, 2, 0, 1, -1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. operator [a,b]=[1, 2] means a*u_x+b*u_y; source=12, x=2, y=0. Give (coefficients,operator,x,y,u_x,u_y,source,operator value−source,satisfied at this point?).
View answer and explanation
([12, 2, 0, 1, -1, 2], [1, 2], 2, 0, 6, -2, 12, -10, No)
Compute both exact partials and the signed residual at the requested point. A zero point residual does not prove a global solution. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [1, 2], 2, 0, 6, -2, 12, -10, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-first-order-point-residual · candidate
26. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-12, -2, -3, 1, -1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, -2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-12, x=-2, y=-3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-12, -2, -3, 1, -1, -2], [1, -2, 1], -2, -3, 2, -1, -4, -12, 12, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -2], [1, -2, 1], -2, -3, 2, -1, -4, -12, 12, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
27. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-11, -1, -2, 2, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, -1, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-11, x=-1, y=-2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-11, -1, -2, 2, 0, -1], [2, -1, 2], -1, -2, 4, 0, -2, -11, 15, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 2, 0, -1], [2, -1, 2], -1, -2, 4, 0, -2, -11, 15, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
28. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-10, 0, -1, 3, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, 0, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-10, x=0, y=-1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-10, 0, -1, 3, 1, 0], [3, 0, 3], 0, -1, 6, 1, 0, -10, 28, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 3, 1, 0], [3, 0, 3], 0, -1, 6, 1, 0, -10, 28, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
29. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-9, 1, 0, 4, -1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 1, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-9, x=1, y=0. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-9, 1, 0, 4, -1, 1], [1, 1, 4], 1, 0, 8, -1, 2, -9, 24, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 4, -1, 1], [1, 1, 4], 1, 0, 8, -1, 2, -9, 24, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
30. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-8, 2, 1, 1, 0, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, 2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-8, x=2, y=1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-8, 2, 1, 1, 0, 2], [2, 2, 1], 2, 1, 2, 0, 4, -8, 16, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, 2], [2, 2, 1], 2, 1, 2, 0, 4, -8, 16, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
31. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-7, -2, 2, 2, 1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, -2, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-7, x=-2, y=2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-7, -2, 2, 2, 1, -2], [3, -2, 2], -2, 2, 4, 1, -4, -7, 9, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 2, 1, -2], [3, -2, 2], -2, 2, 4, 1, -4, -7, 9, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
32. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-6, -1, 3, 3, -1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, -1, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-6, x=-1, y=3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-6, -1, 3, 3, -1, -1], [1, -1, 3], -1, 3, 6, -1, -2, -6, 7, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 3, -1, -1], [1, -1, 3], -1, 3, 6, -1, -2, -6, 7, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
33. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-5, 0, -3, 4, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, 0, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-5, x=0, y=-3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-5, 0, -3, 4, 0, 0], [2, 0, 4], 0, -3, 8, 0, 0, -5, 21, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 4, 0, 0], [2, 0, 4], 0, -3, 8, 0, 0, -5, 21, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
34. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-4, 1, -2, 1, 1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, 1, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-4, x=1, y=-2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-4, 1, -2, 1, 1, 1], [3, 1, 1], 1, -2, 2, 1, 2, -4, 13, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, 1], [3, 1, 1], 1, -2, 2, 1, 2, -4, 13, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
35. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-3, 2, -1, 2, -1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 2, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-3, x=2, y=-1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-3, 2, -1, 2, -1, 2], [1, 2, 2], 2, -1, 4, -1, 4, -3, 13, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 2, -1, 2], [1, 2, 2], 2, -1, 4, -1, 4, -3, 13, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
36. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-2, -2, 0, 3, 0, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, -2, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-2, x=-2, y=0. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-2, -2, 0, 3, 0, -2], [2, -2, 3], -2, 0, 6, 0, -4, -2, 2, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 3, 0, -2], [2, -2, 3], -2, 0, 6, 0, -4, -2, 2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
37. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 1, 4, 1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, -1, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=-1, x=-1, y=1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([-1, -1, 1, 4, 1, -1], [3, -1, 4], -1, 1, 8, 1, -2, -1, 16, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 4, 1, -1], [3, -1, 4], -1, 1, 8, 1, -2, -1, 16, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
38. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, -1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 0, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=0, x=0, y=2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, 0], [1, 0, 1], 0, 2, 2, -1, 0, 0, 2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
39. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[1, 1, 3, 2, 0, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, 1, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=1, x=1, y=3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([1, 1, 3, 2, 0, 1], [2, 1, 2], 1, 3, 4, 0, 2, 1, 11, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 2, 0, 1], [2, 1, 2], 1, 3, 4, 0, 2, 1, 11, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
40. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[2, 2, -3, 3, 1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, 2, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=2, x=2, y=-3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([2, 2, -3, 3, 1, 2], [3, 2, 3], 2, -3, 6, 1, 4, 2, 30, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 3, 1, 2], [3, 2, 3], 2, -3, 6, 1, 4, 2, 30, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
41. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[3, -2, -2, 4, -1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, -2, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=3, x=-2, y=-2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([3, -2, -2, 4, -1, -2], [1, -2, 4], -2, -2, 8, -1, -4, 3, -9, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 4, -1, -2], [1, -2, 4], -2, -2, 8, -1, -4, 3, -9, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
42. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[4, -1, -1, 1, 0, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, -1, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=4, x=-1, y=-1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([4, -1, -1, 1, 0, -1], [2, -1, 1], -1, -1, 2, 0, -2, 4, -2, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [2, -1, 1], -1, -1, 2, 0, -2, 4, -2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
43. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[5, 0, 0, 2, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, 0, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=5, x=0, y=0. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([5, 0, 0, 2, 1, 0], [3, 0, 2], 0, 0, 4, 1, 0, 5, 7, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 2, 1, 0], [3, 0, 2], 0, 0, 4, 1, 0, 5, 7, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
44. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[6, 1, 1, 3, -1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 1, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=6, x=1, y=1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([6, 1, 1, 3, -1, 1], [1, 1, 3], 1, 1, 6, -1, 2, 6, 5, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 3, -1, 1], [1, 1, 3], 1, 1, 6, -1, 2, 6, 5, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
45. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[7, 2, 2, 4, 0, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, 2, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=7, x=2, y=2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([7, 2, 2, 4, 0, 2], [2, 2, 4], 2, 2, 8, 0, 4, 7, 25, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 4, 0, 2], [2, 2, 4], 2, 2, 8, 0, 4, 7, 25, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
46. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[8, -2, 3, 1, 1, -2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, -2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=8, x=-2, y=3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([8, -2, 3, 1, 1, -2], [3, -2, 1], -2, 3, 2, 1, -4, 8, -8, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -2], [3, -2, 1], -2, 3, 2, 1, -4, 8, -8, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
47. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[9, -1, -3, 2, -1, -1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, -1, 2] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=9, x=-1, y=-3. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([9, -1, -3, 2, -1, -1], [1, -1, 2], -1, -3, 4, -1, -2, 9, -8, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 2, -1, -1], [1, -1, 2], -1, -3, 4, -1, -2, 9, -8, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
48. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[10, 0, -2, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[2, 0, 3] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=10, x=0, y=-2. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([10, 0, -2, 3, 0, 0], [2, 0, 3], 0, -2, 6, 0, 0, 10, 2, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 3, 0, 0], [2, 0, 3], 0, -2, 6, 0, 0, 10, 2, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
49. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[11, 1, -1, 4, 1, 1] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[3, 1, 4] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=11, x=1, y=-1. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([11, 1, -1, 4, 1, 1], [3, 1, 4], 1, -1, 8, 1, 2, 11, 22, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 4, 1, 1], [3, 1, 4], 1, -1, 8, 1, 2, 11, 22, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
50. Second-Order Residual with an Explicit Mixed-Term Convention
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[12, 2, 0, 1, -1, 2] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. principal [a,b,c]=[1, 2, 1] means a*u_xx+b*u_xy+c*u_yy, not 2*b*u_xy. source=12, x=2, y=0. Give (coefficients,principal,x,y,u_xx,u_xy,u_yy,source,signed residual,satisfied at the point?).
View answer and explanation
([12, 2, 0, 1, -1, 2], [1, 2, 1], 2, 0, 2, -1, 4, 12, -8, No)
Use u_xx=2*c_xx, u_xy=c_xy and u_yy=2*c_yy in this polynomial model. The requested result is a residual audit, not a general PDE solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, 2], [1, 2, 1], 2, 0, 2, -1, 4, 12, -8, No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-second-order-point-residual · candidate
51. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, -12, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, -12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, -12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
52. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, -11, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, -11, 2], a*u_xx+b*u_xy+c*u_yy, 105, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, -11, 2], a*u_xx+b*u_xy+c*u_yy, 105, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
53. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, -10, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, -10, 3], a*u_xx+b*u_xy+c*u_yy, 64, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -10, 3], a*u_xx+b*u_xy+c*u_yy, 64, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
54. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, -9, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, -9, 4], a*u_xx+b*u_xy+c*u_yy, 65, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, -9, 4], a*u_xx+b*u_xy+c*u_yy, 65, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
55. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, -8, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, -8, 1], a*u_xx+b*u_xy+c*u_yy, 56, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, -8, 1], a*u_xx+b*u_xy+c*u_yy, 56, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
56. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, -7, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, -7, 2], a*u_xx+b*u_xy+c*u_yy, 25, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -7, 2], a*u_xx+b*u_xy+c*u_yy, 25, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
57. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, -6, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, -6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
58. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, -5, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, -5, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, -5, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
59. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, -4, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, -4, 1], a*u_xx+b*u_xy+c*u_yy, 4, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -4, 1], a*u_xx+b*u_xy+c*u_yy, 4, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
60. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, -3, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, -3, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
61. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, -2, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, -2, 3], a*u_xx+b*u_xy+c*u_yy, -20, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, -2, 3], a*u_xx+b*u_xy+c*u_yy, -20, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
62. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, -1, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, -1, 4], a*u_xx+b*u_xy+c*u_yy, -47, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 4], a*u_xx+b*u_xy+c*u_yy, -47, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
63. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, 0, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, 0, 1], a*u_xx+b*u_xy+c*u_yy, -4, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1], a*u_xx+b*u_xy+c*u_yy, -4, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
64. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, 1, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, 1, 2], a*u_xx+b*u_xy+c*u_yy, -15, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 2], a*u_xx+b*u_xy+c*u_yy, -15, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
65. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, 2, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, 2, 3], a*u_xx+b*u_xy+c*u_yy, -32, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 3], a*u_xx+b*u_xy+c*u_yy, -32, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
66. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, 3, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, 3, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 4], a*u_xx+b*u_xy+c*u_yy, -7, elliptic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
67. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, 4, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, 4, 1], a*u_xx+b*u_xy+c*u_yy, 8, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4, 1], a*u_xx+b*u_xy+c*u_yy, 8, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
68. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, 5, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, 5, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 5, 2], a*u_xx+b*u_xy+c*u_yy, 1, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
69. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, 6, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, 6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 6, 3], a*u_xx+b*u_xy+c*u_yy, 24, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
70. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, 7, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, 7, 4], a*u_xx+b*u_xy+c*u_yy, 17, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 7, 4], a*u_xx+b*u_xy+c*u_yy, 17, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
71. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, 8, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, 8, 1], a*u_xx+b*u_xy+c*u_yy, 52, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 8, 1], a*u_xx+b*u_xy+c*u_yy, 52, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
72. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, 9, 2] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, 9, 2], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, 2], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
73. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[2, 10, 3] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([2, 10, 3], a*u_xx+b*u_xy+c*u_yy, 76, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, 3], a*u_xx+b*u_xy+c*u_yy, 76, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
74. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[3, 11, 4] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([3, 11, 4], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, 4], a*u_xx+b*u_xy+c*u_yy, 73, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
75. Classify Only after Fixing b*u_xy rather than 2*b*u_xy
Use the nonzero second-order principal part [a,b,c]=[1, 12, 1] with convention a*u_xx+b*u_xy+c*u_yy. Give (principal,that exact convention string,discriminant b²−4*a*c,classification). Positive, negative and zero discriminants mean hyperbolic, elliptic and parabolic respectively. The all-zero principal part is excluded.
View answer and explanation
([1, 12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic)
Keep the mixed-term convention fixed when computing the discriminant; classification alone proves no existence, uniqueness or well-posedness. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 12, 1], a*u_xx+b*u_xy+c*u_yy, 140, hyperbolic).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-principal-classification · candidate
76. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-12, -2, 2, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-12, -2, 2, 1, 1, 0], 1, [0, 1, 0], No)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, 2, 1, 1, 0], 1, [0, 1, 0], No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
77. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-11, -1, 8, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-11, -1, 8, 2, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, 8, 2, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
78. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-10, 0, 18, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-10, 0, 18, 3, 0, 0], 3, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, 18, 3, 0, 0], 3, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
79. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-9, 1, 8, 1, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-9, 1, 8, 1, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 8, 1, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
80. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-8, 2, 4, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-8, 2, 4, 2, 0, 0], 1, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 4, 2, 0, 0], 1, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
81. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-7, -2, 12, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-7, -2, 12, 3, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 12, 3, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
82. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-6, -1, 6, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-6, -1, 6, 1, 1, 0], 3, [0, 1, 0], No)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 6, 1, 1, 0], 3, [0, 1, 0], No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
83. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-5, 0, 16, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-5, 0, 16, 2, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, 16, 2, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
84. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-4, 1, 6, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-4, 1, 6, 3, 0, 0], 1, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, 6, 3, 0, 0], 1, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
85. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-3, 2, 4, 1, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-3, 2, 4, 1, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, 4, 1, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
86. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-2, -2, 12, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-2, -2, 12, 2, 0, 0], 3, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 12, 2, 0, 0], 3, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
87. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 24, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
88. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[0, 0, 2, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([0, 0, 2, 1, 1, 0], 1, [0, 1, 0], No)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, 1, 0], 1, [0, 1, 0], No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
89. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[1, 1, 8, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([1, 1, 8, 2, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 8, 2, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
90. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[2, 2, 18, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([2, 2, 18, 3, 0, 0], 3, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 18, 3, 0, 0], 3, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
91. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[3, -2, 8, 1, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([3, -2, 8, 1, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, 8, 1, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
92. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[4, -1, 4, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([4, -1, 4, 2, 0, 0], 1, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, 4, 2, 0, 0], 1, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
93. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[5, 0, 12, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([5, 0, 12, 3, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 12, 3, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
94. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[6, 1, 6, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([6, 1, 6, 1, 1, 0], 3, [0, 1, 0], No)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 6, 1, 1, 0], 3, [0, 1, 0], No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
95. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[7, 2, 16, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([7, 2, 16, 2, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 16, 2, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
96. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[8, -2, 6, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([8, -2, 6, 3, 0, 0], 1, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 6, 3, 0, 0], 1, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
97. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[9, -1, 4, 1, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=2. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([9, -1, 4, 1, 0, 0], 2, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, 4, 1, 0, 0], 2, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
98. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[10, 0, 12, 2, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=3. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([10, 0, 12, 2, 0, 0], 3, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, 12, 2, 0, 0], 3, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
99. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[11, 1, 24, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([11, 1, 24, 3, 0, 0], 4, [0, 0, 0], Yes)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, 24, 3, 0, 0], 4, [0, 0, 0], Yes).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
100. A Symbolic Heat Residual Identity, Not a Point Sample
Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[12, 2, 2, 1, 1, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=1. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].
View answer and explanation
([12, 2, 2, 1, 1, 0], 1, [0, 1, 0], No)
A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 2, 1, 1, 0], 1, [0, 1, 0], No).
Reviewed step-by-step derivation
Partial differential equation · qf-mod24-pde-heat-polynomial-identity · candidate
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