Question 11621: 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=[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?).

• G001 — 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?).

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=[8, -2, 3, 1, 1, -2];basis=[1,x,y,x^2,xy,y^2]

u(x,y)=(8)+(-2)*x+(3)*y+(1)*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(8, 9, -18)

PDE x slice constant coefficient = -1

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=[8, -2, 3, 1, 1, -2]

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

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

g(-2+h)-g(-2)=(1)*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(1, -4)

PDE x partial derivative value = -3

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

difference quotient=-3+(1)*h

PDE x partial:quotient=-3+(1)*h;epsilon>0;delta=epsilon/1

0<abs(h)<delta gives abs(quotient-(-3))=abs(1)*abs(h)<epsilon;limit=-3

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(8, 4, 4)

PDE y slice constant coefficient = 16

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=[8, -2, 3, 1, 1, -2]

ascending y slice coefficients=[16, 1, -2]

PDE y partial:g(t)=(16)+(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 = -11

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

difference quotient=-11+(-2)*h

PDE y partial:quotient=-11+(-2)*h;epsilon>0;delta=epsilon/2

0<abs(h)<delta gives abs(quotient-(-11))=abs(-2)*abs(h)<epsilon;limit=-11

first order term0 := mul(3, -3)

first order term0 = -9

first order term1 := mul(-2, -11)

first order term1 = 22

first order operator sum := sum(-9, 22)

first order operator sum = 13

first order signed residual := sub(13, 8)

first order signed residual = 5

query point=[-2, 3];operator value=13;source=8;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

([8, -2, 3, 1, 1, -2], [3, -2], -2, 3, -3, -11, 8, 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.