This is the question my softmore differential equations professor asked on our practice exam:
Show that the substitution $y=ux$ in the first-order differential equation $$p(x,y)\;dx+q(x,y)\;dy=0$$ results in an ODE (in u and x) which can be solved by "separation of variables" if p and q are homogeneous of the same degree (meaning $p(tx,ty)=t^dp(x,y)$ and $q(yx,ty)=t^dp(x,y)$ for some integer d and all real $t\neq0$.)
What is the expression you get for the solution of the original ODE (i.e., after undoing the substitution). Check that it works.
Note: you'll need to have the correct substitution for dy here for things to work.
This is how far I've gotten
Where should I go from here? I am completely lost. HELP!