Re: [理工] [工數]-高階ODE
※ 引述《winer8 (快來明星3 缺1 )》之銘言：
: (x^2-2x)y''+2(1-x)y'+2y=6(x^2-2x)^2
: 我看出一齊性解y=x^2 let y=vx^2 帶入
: (x^2-2x)(v''^2+2v'x+2v)+2(1-x)(2xv+v'x^2)+2vx^2=6(x^2-2x)^2
: v"(x^4-2x^3)+v'(-2x)=6(x^2-2x)^2
: v"+v'(-2/(x^2-2x))=6(x-2)^2
: v"+v'(-2/(x^2-2x))=6(x-2)^2
: I=x/(x-2) v'=2x^2(x-2)-6x(x-2)+C1(x-2)/x
: v=1/2 x^4-10/3 x^3+6x^2+C1x-2C1lnx+C2
: y=vx^2
: 但答案是=y=c1x^2+c2(x-1)+(x^3-3x^2)(x-1) 不知道是錯在哪呢?
: 還有x^3y"-4xy'+4y=0 這題完全沒頭緒
: 答暗是 y=cix+c2xexp(-4/x)
: 請教各位了 謝謝
2(1-x) 2
y''+ -------- y'+ -------- y = 6x(x-2)
x(x-2) x(x-2)
y=uv u=x^2 y=vx^2
y'=v'x^2+v2x
y''=v''x^2+v'2x+v'2x+v2 = v''x^2 +4v'x+ 2v
~~~~
4 2(1-x) 6x(x-2)
v''+ ( --- + --------) v' = ---------
x x(x-2) x^2
3 -1 6x(x-2)
v''+ ( --- + ---- ) v' = ---------
x x-2 x^2
x^3
I(x) = -----
x-2
x-2 x^3 6x(x-2)
v'= ------[∫----- ------- dx + c1 ]
x^3 x-2 x^2
x-2
v'= ------[2x^3 + c1]
x^3
x-2
v'= 2(x-2) + c1------
x^3
x-2
v = ∫2(x-2) dx + c1∫------ dx
x^3
-2 1
v = x^2-4x +c1 ∫(----- + ----- )dx +c2
x^3 x^2
1 -1
v = x^2-4x +c1(---- + -----) +c2
x^2 x
y = uv
y = x^4 -4x^3 +c1(1-x) + c2x^2
你在算看看!!
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