{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-partial-differential-equation"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-partial-differential-equation-01","title":{"zh-Hant":"偏微分方程：常係數二階算子的主部分類與二次多項式候選的符號殘差","en":"Partial differential equation: principal-part classification and symbolic residuals of quadratic polynomial candidates"},"summary":{"zh-Hant":"在 ℝ² 上給定線性常係數算子 L(u)=u_xx+2u_xy−3u_yy+u_x 與右端 g(x,y)=2x+y−2。先依明示的主部約定 a u_xx+b u_xy+c u_yy 與判別式 b²−4ac 對 L 分類；再對三個二次多項式候選 u_A=x²+xy+y²、u_B=x²+2xy、u_C=x²+xy+y²+y 精確計算符號殘差 R=L(u)−g，判定哪些候選在全 ℝ² 上滿足 L(u)=g，並說明 u_B 的點殘差在整條直線 y=−8 上為零，為何仍不構成解。","en":"On ℝ², take the linear constant-coefficient operator L(u)=u_xx+2u_xy−3u_yy+u_x and the right-hand side g(x,y)=2x+y−2. First classify L using the explicit principal-part convention a u_xx+b u_xy+c u_yy with discriminant b²−4ac; then, for the three quadratic polynomial candidates u_A=x²+xy+y², u_B=x²+2xy and u_C=x²+xy+y²+y, compute the symbolic residual R=L(u)−g exactly, decide which candidates satisfy L(u)=g on all of ℝ², and explain why u_B is still not a solution although its point residual vanishes on the whole line y=−8."},"mko_ids":["mko-partial-differential-equation"]}],"title":"Concept derivation guides"}