finding a general solution of an ODE

Question

I'm trying to find a general solution for $x(x+y)y'=y(x-y)$ using the substitution method. I literally have no idea where to start with this. almost 2 hours into it and no meaningful progress. I really need some help

Answer 1

Make the ODE explicit $$ y'=\frac{y(x-y)}{x(x+y)} $$ and recognize a homogeneous equation. Set $y=xu$ to get $$ xu'+u=\frac{u(1-u)}{1+u}\iff xu'=-\frac{2u^2}{1+u} $$ which can be separated and integrated.