Reduce a differential equation with a substitution

$y=ux^2$ gives $y^\prime=2ux+x^2u^\prime$. Substitution gives $x^2(2ux+x^2u^\prime)-2x(ux^2)+u^2x^4=x^4(u^\prime+u^2)=0$. Since $x>0$, $u^\prime+u^2=0$.

For $u\ne0$, $-u^{-2}\,du=dx$, so $1/u=x+C$. Returning to $y$ gives $y=x^2/(x+C)$ on intervals with $x>0$ and $x+C\ne0$.

$u=0$ also solves the reduced equation, giving $y=0$ in the original equation. This valid solution is omitted by division by $u^2$ and is not represented by a finite value of $C$ in the printed family.