数学物理方程试题 Laplace变换表 原函数f(t) P1(x)= (n!)2(-n)!(2 (2+1)xP2(x)=(l+1)P1+1(x)+lP1-1(x) erfc P+1(x)=P(x)+(l+1)P1(x) P√P P+1(x)=(2l+1)P4(x)+P-1(x) P(x)=Pl+1(x)-2rP(x)+P-1(x) J(x)= -2aVt 2/Perf kA(k++1(2 注意:除第一题应当直接写出答案外, 其余各题均须写出必要的关键步骤. (20分)写出下列各本征值问题的解: y"(x)+My(x)=0 y(0)=0,y(l)=0; y(x)+Ay(x)=0, y(0)=0,y(l)=0 y"(x)+My(x)=0, y(x)=y(x+2丌)=0✄ ☎ ✆ ✝ ✞ ✟ ➔ → ✠ ✡ ☛ Pl(x) = X l n=0 1 (n!)2 (l + n)! (l − n)!  x − 1 2 n (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl−1(x) P 0 l+1(x) = xP 0 l (x) + (l + 1)Pl(x) P 0 l+1(x) = (2l + 1)Pl(x) + P0 l−1 (x) Pl(x) = P0 l+1(x) − 2xP 0 l (x) + P0 l−1 (x) Jν(x) = X∞ k=0 (−) k k!Γ (k + ν + 1) x 2 2k+ν Laplace ➑➒☛ ✾ ✹✸ f(t) ➓ ✹✸ F(p) 1 1 p erfc α 2 √ t 1 p e −α √p 1 √ πα sin 2√ αt 1 p √ p e −α/p 1 √ πt cos 2√ αt 1 √ p e −α/p 1 √ πt e −α 2/4t 1 √ p e −α √p 1 √ πt e −2α √ t 1 √p e −α 2/perfc α √p ☞✌❀ ✏✑✒✓✍✎✖✗✦✧✙✏✚✛ ✜✢✣✓✤✑✦✧★✔✩✪✫✬✭❉ ✱✲ (20 ✶) ✒ ➠ ➋➌✓✔✕✼✖✵P▼ ❀ (1)    y 00(x) + λy(x) = 0, y(0) = 0, y(l) = 0; (2)    y 00(x) + λy(x) = 0, y 0 (0) = 0, y0 (l) = 0; (3)    y 00(x) + λy(x) = 0, y(x) = y(x + 2π) = 0; 10