作者kagato (包)
看板Grad-ProbAsk
标题Re: [理工] [工数]-ODE
时间Thu Jan 21 11:14:13 2010
※ 引述《lalala419 (啦啦啦)》之铭言：
: 1.
: (xlny-xlnx)dy=(ylny-ylnx-x)dx
: 这能用正合作吗? 周易的观察法我不熟不太会用
: 2.
: y'+(1/3)y=(1/3)(1-2x)y*4
: 我同除y四方开始算
: 令z=1/y*3
1.
(ylny-ylnx-x)dx - (xlny-xlnx)dy = 0
let M = ylny-ylnx-x ; N = - (xlny-xlnx)
My-Nx = lny + 1 - lnx +lny - lnx - 1 = 2(lny-lnx)
I = exp^{∫(My-Nx)/Ndx} = 1/x^2
→ (ylny-ylnx-x)/x^2dx - (lny-lnx)/xdy = 0
let ψ=c为解
ψ(x,y) = y(lnx)/x-y*(ln(y)-1)/x + f(x) _____(1)
ψ(x,y) = y(lnx)/x-y*(ln(y)-1)/x -lnx + g(y) _____(2)
由1,2 → ψ = y(lnx)/x-y*(ln(y)-1)/x - lnx = c
______________________________
2. y'+(1/3)y=(1/3)(1-2x)y*4
let t = y^-3 , t' = -3y^(-4)y'
→ 3y^(-4)y' + y^(-3) = 1 - 2x
-t' + t = 1 - 2x ; t' - t = 2x - 1
t*exp^(-x) = ∫(2x-1)*exp^(-x)dx + c
t = -2x - 2 +1 + c*exp^x
→ y^(-3) = -2x - 2 +1 + c*exp^x
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