Re: [理工] [工數]-高階ODE
2 2 2
(x +1)y'' - 2xy' +2y= 6(x +1)
得x=Yh後
let y=xz
y'=z+xz'
y''=2z'+xz''
(x^2 + 1)(2z'+xz'')-2x(z+xz')+2(xz)=6(x^2+1)^2
(x^2+1)xz''+(2)z'=6(x^2+1)^2
let z'=u
z''=u'
2 6(x^2+1)
u'+--------u=--------- -----(1)
(x^2+1)x x
-2x 2
∫ ----- +--- dx
I= e^ x^2+1 x
x^2
I=e^(2lnx-lnx^2+1)=------
x^2+1
(1)xI
=> d(uI)=6x
uI=3x^2+C
x^2+1
u=3(x^2+1)+C----- = z'
x^2
-1
z=x^3+3x+C(x+---)+D
x
y=xz=x^4+3x^2+C(x^2-1)+Dx
--
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07/25 20:17, , 1F
07/25 20:17, 1F
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