作者finalgod (0_o安ㄚ)
看板trans_math
标题Re: Frobenius~
时间Wed Feb 15 02:58:11 2006
※ 引述《amymayyam (考试挂网)》之铭言：
: xy"+2(1-x)y'+(x-2)y=0
: 麻烦高手指导!!
: 急~~~~~
∞ ∞ ∞
y=ΣC[m]X^(m+r)，y'=Σ(m+r)C[m]X^(m+r-1)，y''=Σ(m+r)(m+r-1)C[m]X^(m+r-2)
m=0 m=0 m=0
xy"+2(1-x)y'+(x-2)y=0
∞ ∞ ∞
xΣ(m+r)(m+r-1)C[m]X^(m+r-2)+2(1-x)Σ(m+r)C[m]X^(m+r-1)+(x-2)ΣC[m]X^(m+r)=0
m=0 m=0 m=0
∞ ∞ ∞
Σ(m+r)(m+r-1)C[m]X^(m+r-1)+Σ2(m+r)C[m]X^(m+r-1)-Σ2(m+r)C[m]X^(m+r)
m=0 m=0 m=0
∞ ∞
+ΣC[m]X^(m+r+1)-Σ2C[m]X^(m+r)=0
m=0 m=0
∞ ∞ ∞
Σ(m+r)(m+r+1)C[m]X^(m+r-1)-Σ2(m+r+1)C[m]X^(m+r)+ΣC[m]X^(m+r+1)=0
m=0 m=0 m=0
r(r+1)C[0]X^(r-1)+{(r+1)(r+2)C[1]-2(r+1)C[0]}X^r
∞
+Σ{(s+r+2)(s+r+3)C[s+2]-2(s+r+2)C[s+1]+C[s]}X^(s+r+1)=0
s=0
2C[0]
C[1]= -----
(r+2)
2C[s+1] C[s]
C[s+2]= ------- - --------------
(s+r+3) (s+r+2)(s+r+3)
3C[0] 4C[0] 5C[0]
C[2]= ----------，C[3]= ---------------，C[4]= --------------------
(r+2)(r+3) (r+2)(r+3)(r+4) (r+2)(r+3)(r+4)(r+5)
(n+1)C[0] (n+1)(r+1)!
C[n]= ---------------------------- = ----------- C[0]
(r+2)(r+3)．．．(r+n)(r+n+1) (r+n+1)!
取r=0，得C[n]=(1/n!)C[0]，则C[1]=C[0]，C[2]=(1/2!)C[0]，C[3]=(1/3!)C[0]．．．
y1={1+x+(1/2!)x^2+(1/3!)x^3+(1/4!)x^4+．．．(1/n!)x^n+．．．}= e^x
e^[-2∫{(1-x)/x}dx] e^[-2ln[x]+2x] (1/x^2)e^2x
y2=e^x∫--------------------dx=e^x∫--------------dx=e^x∫-----------dx
(e^x)^2 e^2x e^2x
1
=e^x∫---dx= - e^x/x
x^2
General Solution：y=C[1]e^x+C[2]e^x/x
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◆ From: 140.122.230.117
※ 编辑: finalgod 来自: 140.122.230.117 (02/16 20:22)
※ 编辑: finalgod 来自: 140.122.230.117 (03/13 23:41)