Re: [理工] [工數]-高階ODE

[iyenn (曉風)]

  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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