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R0022-11679 · q-20261013-advance-20261024-r06-partial-differential-equation-0029
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?).
G001 — 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?).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.
Before
original coefficients=[-9, 1, 0, 4, -1, 1];basis=[1,x,y,x^2,xy,y^2]
After
u(x,y)=(-9)+(1)*x+(0)*y+(4)*x^2+(-1)*x*y+(1)*y^2
Conditions: none
Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic.
Rule: bivariate_scalar_polynomial_basis_roles · Uses: G001
Before
u_xx constant := mul(2, 4)
After
u_xx constant = 8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
u_yy constant := mul(2, 1)
After
u_yy constant = 2
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
u coefficients=[-9, 1, 0, 4, -1, 1];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y
After
u_xx=8;u_xy=-1;u_yy=2
Conditions: none
Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem.
Rule: polynomial_second_partials_explicit_monomials · Uses: G001, S003
Before
principal=[1, 1, 4];point=[1, 0]
After
operator=a*u_xx+b*u_xy+c*u_yy;not 2*b*u_xy
Conditions: none
The mixed term is b*u_xy,not the alternative convention2b*u_xy.Retain the query point roles even though these second derivatives are constant for this polynomial.
Rule: second_order_declared_mixed_term · Uses: G001, S004
Before
second order term0 := mul(1, 8)
After
second order term0 = 8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
second order term1 := mul(1, -1)
After
second order term1 = -1
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
second order term2 := mul(4, 2)
After
second order term2 = 8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
second order operator sum := sum(8, -1, 8)
After
second order operator sum = 15
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
second order signed residual := sub(15, -9)
After
second order signed residual = 24
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
query point=[1, 0];operator value=15;source=-9;signed residual=24
After
satisfied at declared point=No;no general solution claim
Conditions: none
This flag tests the residual at the declared point.A point pass does not generally prove a global solution or existence,uniqueness,regularity or well-posedness.A global identity requires a separate full symbolic residual audit.
Rule: PDE_point_obligation_not_global_solution · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute second order signed residual with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute second order operator sum with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute second order term2 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute second order term1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute second order term0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute u_yy constant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute u_xx constant 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.([-9, 1, 0, 4, -1, 1], [1, 1, 4], 1, 0, 8, -1, 2, -9, 24, No)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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