方程y'^3+2xy'-y=0的通解
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方程y'^3+2xy'-y=0的通解
![方程y'^3+2xy'-y=0的通解](/uploads/image/z/18887233-49-3.jpg?t=%E6%96%B9%E7%A8%8By%27%5E3%2B2xy%27-y%3D0%E7%9A%84%E9%80%9A%E8%A7%A3)
∵令y'=p,则代入原方程得y=p³+2xp.(1)
==>p=3p²p'+2p+2xp' (两端对x求导数)
==>3p²dp+2xdp+pdx=0
==>3p³dp+2xpdp+p²dx=0 (两端同乘p)
==>3d(p^4)+4d(xp²)=0
==>3p^4+4xp²=C (C是积分常数)
==>x=C/p²-3p²
==>y=2C/p-5p³
∴原方程的通解参数形式是x=C/p²-3p²,y=2C/p-5p³ (C是积分常数,p是参数).
==>p=3p²p'+2p+2xp' (两端对x求导数)
==>3p²dp+2xdp+pdx=0
==>3p³dp+2xpdp+p²dx=0 (两端同乘p)
==>3d(p^4)+4d(xp²)=0
==>3p^4+4xp²=C (C是积分常数)
==>x=C/p²-3p²
==>y=2C/p-5p³
∴原方程的通解参数形式是x=C/p²-3p²,y=2C/p-5p³ (C是积分常数,p是参数).
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