Solve the given initial$-$value problem. The DE is homogeneous. xy$^2$dy$/$dx$=$ y$^3$ x$^3,$ y$(1)=1$

The strategy to find a solution to a homogenous differential equation is to make a substitution that results in a separable equation. For a homogenous equation, either of the substitutions

u $=$yx

or

v $=$ xy$.$

will make this work. Let

u $=$yx$,$

or

y $=$ ux$.$

Then

dy $=$ u dx $+$ x du

by the product rule. Use this substitution to simplify the given equation.

xy$2$ dy $+($y$3+$ x$3)$ dx$=0$

xy$2$ dy $$ y$3$ dx $+$ x$3$ dx$=0$