Re: [理工] [微方]一階ODE
1.lny=v, y=e^v, y'=e^v*dv/dx
代入得
x*e^v*dv/dx-4*x^2*e^v+2*v*e^v=0
消去e^v
x*dv/dx-4*x^2+2*v=0
v'+2/x*v=4*x
一階線性微方
I=exp(積分2/xdx)=e^(2*lnx)=x^2
vI=積分4x*x^2dx=x^4+c
v=x^2+c*x^-2=lny
y=e^(x^2+c*x^-2)
2.
y*(ydx+3xdy)-x(9ydx+6xdy)=0
y*d(xy^3)/(y^2)-x(d(x^9y^6)/(x^8y^5))=0
同乘x^8y^5
x^8y^4d(xy^3)-xd(x^9y^6)=0
x^7y^4d(xy^3)-d(x^9y^6)=0
寫成(xy^3)^md(xy^3)-(x^9y^6)^nd(x^9y^6)=0
x^my^3mx^-9ny^-6nd(xy^3)-d(x^9y^6)=0
可得m-9n=7 3m-6n=4
解m,n=-2/7, -17/21
(xy^3)^(-2/7)d(xy^3)-(x^9y^6)^(-17/21)d(x^9y^6)=0
積分
7/5(xy^3)^5/7=21/4(x^9y^6)^4/21+c
3.y'=(4x^2+y^2)/xy=4x/y+y/x
令y/x=v
y'=v+xdv/dx
v+xdv/dx+4/v+v
dv/dx=4/xv
v*dv/dx=4/x
vdv=4dx/x
積分
v^2/2=4*lnx+c
y^2=8*x^2*lnx+cx^2
根據初始條件知c=4
※ 引述《dkcheng (電磁霸主)》之銘言：
: 1. Solve xy'-4x^2y+2ylny=0 by let v=lny
: 2. Solve (y^2-9xy)dx+(3xy-6x^2)dy=0
: 3. Solve y'=(4x^2+y^2)/xy , y(1)=-2
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 111.111.111.111
※ 編輯: ocean5566 (61.60.206.77), 06/02/2017 01:10:28
討論串 (同標題文章)