Question 11625: 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=[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?).

• G001 — 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?).

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=[12, 2, 0, 1, -1, 2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(12)+(2)*x+(0)*y+(1)*x^2+(-1)*x*y+(2)*y^2

PDE x slice fixed coordinate squared := mul(0, 0)

PDE x slice fixed coordinate squared = 0

PDE x slice fixed linear term := mul(0, 0)

PDE x slice fixed linear term = 0

PDE x slice fixed square term := mul(2, 0)

PDE x slice fixed square term = 0

PDE x slice constant coefficient := sum(12, 0, 0)

PDE x slice constant coefficient = 12

PDE x slice cross contribution := mul(-1, 0)

PDE x slice cross contribution = 0

PDE x slice linear coefficient := sum(2, 0)

PDE x slice linear coefficient = 2

PDE x slice:vary x;fix y=0;original coefficients=[12, 2, 0, 1, -1, 2]

ascending x slice coefficients=[12, 2, 1]

PDE x partial:g(t)=(12)+(2)*t+(1)*t^2;point=2;h!=0

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

PDE x partial twice quadratic coefficient := mul(2, 1)

PDE x partial twice quadratic coefficient = 2

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

PDE x partial derivative point term = 4

PDE x partial derivative value := sum(2, 4)

PDE x partial derivative value = 6

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

difference quotient=6+(1)*h

PDE x partial:quotient=6+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(6))=abs(1)*abs(h)<epsilon;limit=6

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(1, 4)

PDE y slice fixed square term = 4

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

PDE y slice constant coefficient = 20

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

PDE y slice cross contribution = -2

PDE y slice linear coefficient := sum(0, -2)

PDE y slice linear coefficient = -2

PDE y slice:vary y;fix x=2;original coefficients=[12, 2, 0, 1, -1, 2]

ascending y slice coefficients=[20, -2, 2]

PDE y partial:g(t)=(20)+(-2)*t+(2)*t^2;point=0;h!=0

g(0+h)-g(0)=(-2)*h+(2)*(2*(0)*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, 0)

PDE y partial derivative point term = 0

PDE y partial derivative value := sum(-2, 0)

PDE y partial derivative value = -2

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

difference quotient=-2+(2)*h

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

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

first order term0 := mul(1, 6)

first order term0 = 6

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

first order term1 = -4

first order operator sum := sum(6, -4)

first order operator sum = 2

first order signed residual := sub(2, 12)

first order signed residual = -10

query point=[2, 0];operator value=2;source=12;signed residual=-10

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

([12, 2, 0, 1, -1, 2], [1, 2], 2, 0, 6, -2, 12, -10, 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.