[工數] legendre equation...

[momentaneous (請給我一個微笑)]

想請問一下這一題證明要怎麼證...

我算到一半都會卡卡的><


Consider a Legendre equation

(1-x^2)y''-2xy'+n(n+1)y=0



試證:Q2(x)=1/4(3x^2-1)ln(1+x/1-x)-(3/2)x


P2(x)=1/2(3x^2-1)


                          1
Q2(x)=-P2(x) ∫ --------------------  dx
                 [P2(x)]^2(1-x^2)




可以請會的人幫忙解一下嗎...


感激不盡











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◆ From: 140.135.43.92
※ 編輯: momentaneous    來自: 140.135.43.92        (01/07 04:25)
→ mzhrqoc01   :已知P_2（x）為一解，將Q_2(x)=v(x)*P_2(x)帶回去
→ mzhrqoc01   :=> (1-x^2)P'v''+[2P'(1-x^2)-2xP]v'=0
→ mzhrqoc01   :=>(v'/v)=-[2(P'/P)+(-2x/(1-x^2))]
→ mzhrqoc01   :=> ln(v')= -[2 ln(P) + ln(1-x^2) + C ]