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R0022-11654 · q-20261013-advance-20261024-r06-partial-differential-equation-0054
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.
G001 — 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.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
principal mixed coefficient squared := mul(-9, -9)
After
principal mixed coefficient squared = 81
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
Before
principal a times c := mul(1, 4)
After
principal a times c = 4
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
principal four times ac := mul(4, 4)
After
principal four times ac = 16
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
principal discriminant := sub(81, 16)
After
principal discriminant = 65
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, S003
Before
nonzero principal=[1, -9, 4];operator=a*u_xx+b*u_xy+c*u_yy
After
principal symbol=a*xi^2+b*xi*eta+c*eta^2;discriminant=65;classification=hyperbolic
Conditions: none
The symmetric principal-symbol matrix has off-diagonal b/2,hence determinant ac−b²/4.D>0 gives an indefinite symbol,D<0 a definite one,and D=0 a degenerate one,under the declared convention.The all-zero principal part is excluded;classification alone gives no well-posedness.
Rule: principal_symbol_mixed_convention_classification · Uses: G001, S004
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 — Recompute principal discriminant with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute principal four times ac with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute principal a times c with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute principal mixed coefficient squared with exact rational arithmetic, not rounded decimals.([1, -9, 4], a*u_xx+b*u_xy+c*u_yy, 65, hyperbolic)
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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