微分方程(x)^2y'+2xy=3y^2,y(2)=2/9
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微分方程(x)^2y'+2xy=3y^2,y(2)=2/9
![微分方程(x)^2y'+2xy=3y^2,y(2)=2/9](/uploads/image/z/19205436-12-6.jpg?t=%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%28x%29%5E2y%27%2B2xy%3D3y%5E2%2Cy%282%29%3D2%2F9)
方程两边同除以x^2,
dy/dx+2y/x=3(y/x)^2,
设u=y/x,
dy/dx=3u^2-2u
y=ux,
dy/dx=u+xdu/dx,
3u^2-2u=u+xdu/dx,
du/[3u(u-1)]=dx/x,
两边同时积分,
1/3∫[1/(u-1)-1/u]du=ln(Cx),(C是常数)
(1/3)[ln(u-1)/u]=ln(Cx)
(u-1)^(1/3)/u^(1/3)=Cx,
(y/x-1)^(1/3)/(y/x)^(1/3)=Cx,
通解为:[(y-x)/y]^(1/3)=Cx,
当x=2,y=2/9,
(2/9-2)^(1/3)/(2/9)^(1/3)=C*2,
-2=2C,
∴C=-1,
∴当y(2)=2/9特解是:
:[(y-x)/y]^(1/3)=-x.
dy/dx+2y/x=3(y/x)^2,
设u=y/x,
dy/dx=3u^2-2u
y=ux,
dy/dx=u+xdu/dx,
3u^2-2u=u+xdu/dx,
du/[3u(u-1)]=dx/x,
两边同时积分,
1/3∫[1/(u-1)-1/u]du=ln(Cx),(C是常数)
(1/3)[ln(u-1)/u]=ln(Cx)
(u-1)^(1/3)/u^(1/3)=Cx,
(y/x-1)^(1/3)/(y/x)^(1/3)=Cx,
通解为:[(y-x)/y]^(1/3)=Cx,
当x=2,y=2/9,
(2/9-2)^(1/3)/(2/9)^(1/3)=C*2,
-2=2C,
∴C=-1,
∴当y(2)=2/9特解是:
:[(y-x)/y]^(1/3)=-x.