2x^3y'=y(2x^2-y^2)的通解那个y'是在下面的不是和3连在一起的

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2x^3y'=y(2x^2-y^2)的通解
那个y'是在下面的不是和3连在一起的

2x^3/y'=y(2x^2-y^2),2x^3dx=ydy(x^2-y^2),
x^2dx^2=(1/2)dy^2(x^2-y^2).
令 u=x^2,v=y^2,则化为 udu=(1/2)dv(u-v),
即 du/dv=(1/2)(1-v/u) 为齐次方程.
令 u/v=p,则 u=pv,du/dv=v+pdv/dp,
则方程化为 v+pdv/dp=(1/2)(p-1)/p
dv/dp+v/p=(1/2)(p-1)/p^2 为一阶线性微分方程.
v = e^(-∫dp/p){(1/2)∫[(p-1)/p^2]e^(∫dp/p)dp+C}
= (1/p)[(1/2)(p-lnp)+C],
即 u = pv = (1/2)[u/v-ln(u/v)]+C,
即得通解 x^2 =(1/2)[x^2/y^2-ln(x^2/y^2)]+C.

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