I'm trying to solve the non linear differential equation $yy'+xy=x^3$.
Making the substitution $u=\dfrac{x^2}{y}$, solving for $y$ and taking the derivative, gives
$y'=\dfrac{2ux-u'x^2}{u^2}$
With that substitution, the equation turns into a separable one, like this $$u'=\dfrac{-(u^3-u^2-2u)}{x}$$
My question is: Is it correct to make a substitution that is nonlinear for $x$? I ask because i saw in the more of the books that the substitutions are generally like this $y=ux$ or $y=u/x$, so i am not sure.