Question 11615: One-Point Residual of a Declared First-Order PDE Operator

Original question and givens

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?).

• G001 — 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?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-partial-differential-equation

Step-by-step derivation

original coefficients=[2, 2, -3, 3, 1, 2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(2)+(2)*x+(-3)*y+(3)*x^2+(1)*x*y+(2)*y^2

PDE x slice fixed coordinate squared := mul(-3, -3)

PDE x slice fixed coordinate squared = 9

PDE x slice fixed linear term := mul(-3, -3)

PDE x slice fixed linear term = 9

PDE x slice fixed square term := mul(2, 9)

PDE x slice fixed square term = 18

PDE x slice constant coefficient := sum(2, 9, 18)

PDE x slice constant coefficient = 29

PDE x slice cross contribution := mul(1, -3)

PDE x slice cross contribution = -3

PDE x slice linear coefficient := sum(2, -3)

PDE x slice linear coefficient = -1

PDE x slice:vary x;fix y=-3;original coefficients=[2, 2, -3, 3, 1, 2]

ascending x slice coefficients=[29, -1, 3]

PDE x partial:g(t)=(29)+(-1)*t+(3)*t^2;point=2;h!=0

g(2+h)-g(2)=(-1)*h+(3)*(2*(2)*h+h^2)

PDE x partial twice quadratic coefficient := mul(2, 3)

PDE x partial twice quadratic coefficient = 6

PDE x partial derivative point term := mul(6, 2)

PDE x partial derivative point term = 12

PDE x partial derivative value := sum(-1, 12)

PDE x partial derivative value = 11

PDE x partial:h!=0;increment=h*((-1)+2*(3)*(2)+(3)*h)

difference quotient=11+(3)*h

PDE x partial:quotient=11+(3)*h;epsilon>0;delta=epsilon/3

0<abs(h)<delta gives abs(quotient-(11))=abs(3)*abs(h)<epsilon;limit=11

PDE y slice fixed coordinate squared := mul(2, 2)

PDE y slice fixed coordinate squared = 4

PDE y slice fixed linear term := mul(2, 2)

PDE y slice fixed linear term = 4

PDE y slice fixed square term := mul(3, 4)

PDE y slice fixed square term = 12

PDE y slice constant coefficient := sum(2, 4, 12)

PDE y slice constant coefficient = 18

PDE y slice cross contribution := mul(1, 2)

PDE y slice cross contribution = 2

PDE y slice linear coefficient := sum(-3, 2)

PDE y slice linear coefficient = -1

PDE y slice:vary y;fix x=2;original coefficients=[2, 2, -3, 3, 1, 2]

ascending y slice coefficients=[18, -1, 2]

PDE y partial:g(t)=(18)+(-1)*t+(2)*t^2;point=-3;h!=0

g(-3+h)-g(-3)=(-1)*h+(2)*(2*(-3)*h+h^2)

PDE y partial twice quadratic coefficient := mul(2, 2)

PDE y partial twice quadratic coefficient = 4

PDE y partial derivative point term := mul(4, -3)

PDE y partial derivative point term = -12

PDE y partial derivative value := sum(-1, -12)

PDE y partial derivative value = -13

PDE y partial:h!=0;increment=h*((-1)+2*(2)*(-3)+(2)*h)

difference quotient=-13+(2)*h

PDE y partial:quotient=-13+(2)*h;epsilon>0;delta=epsilon/2

0<abs(h)<delta gives abs(quotient-(-13))=abs(2)*abs(h)<epsilon;limit=-13

first order term0 := mul(3, 11)

first order term0 = 33

first order term1 := mul(2, -13)

first order term1 = -26

first order operator sum := sum(33, -26)

first order operator sum = 7

first order signed residual := sub(7, 2)

first order signed residual = 5

query point=[2, -3];operator value=7;source=2;signed residual=5

satisfied at declared point=No;no general solution claim

Reverse checks

• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute first order signed residual with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute first order operator sum with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute first order term1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute first order term0 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute PDE y partial derivative value with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute PDE y partial derivative point term with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute PDE y partial twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute PDE y slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute PDE y slice cross contribution with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute PDE y slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute PDE y slice fixed square term with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute PDE y slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute PDE y slice fixed coordinate squared with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute PDE x partial derivative value with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute PDE x partial derivative point term with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute PDE x partial twice quadratic coefficient with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute PDE x slice linear coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute PDE x slice cross contribution with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute PDE x slice constant coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute PDE x slice fixed square term with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute PDE x slice fixed linear term with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute PDE x slice fixed coordinate squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([2, 2, -3, 3, 1, 2], [3, 2], 2, -3, 11, -13, 2, 5, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.