# Question 11653: Classify Only after Fixing b*u_xy rather than 2*b*u_xy [Return to catalogue page 59](https://ocme.evemisslab.com/en/questions/page/59/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations/#q-20261013-advance-20261024-r06-partial-differential-equation-0053) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0022-11653 · q-20261013-advance-20261024-r06-partial-differential-equation-0053 ## Original question and givens Use the nonzero second-order principal part [a,b,c]=[3, -10, 3] 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]=[3, -10, 3] 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. ## 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](https://ocme.evemisslab.com/en/mko/mko-partial-differential-equation/) ## Step-by-step derivation S001 · calculationBefore principal mixed coefficient squared := mul(-10, -10) After principal mixed coefficient squared = 100 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore principal a times c := mul(3, 3) After principal a times c = 9 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore principal four times ac := mul(4, 9) After principal four times ac = 36 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore principal discriminant := sub(100, 36) After principal discriminant = 64 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore nonzero principal=[3, -10, 3];operator=a*u_xx+b*u_xy+c*u_yy After principal symbol=a*xi^2+b*xi*eta+c*eta^2;discriminant=64;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. Step identifiers and dependenciesRule: principal_symbol_mixed_convention_classification · Uses: G001, S004 ## Reverse checks - 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. ## Result ([3, -10, 3], a*u_xx+b*u_xy+c*u_yy, 64, hyperbolic) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 323313f6b7a36094b806372e49ca9181e26225c29069359abeb9049b1fb1f98c - candidate record SHA-256: eb240b3bd7ae6994a2e6cfe1f037400337a56764b959f212191edbd0f8941c96 - semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac - question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r06-partial-differential-equation-0053.json)