Solution
= Solution
Take $U=Y$ and $V=XY^2$. The inverse is $y=u$, $x=v/u^2$, with absolute Jacobian $u^{-2}$. The original unit-square support becomes $0<u<1$ and $0<v<u^2$, so the joint density is $u^{-2}$ there. For a fixed $0<v<1$, the admissible range is $\sqrt v<u<1$. Therefore
$$
\boxed{f_V(v)=\begin{cases}v^{-1/2}-1,&0<v<1,\\0,&\text{otherwise}.\end{cases}}
$$
It is nonnegative and integrates to $\int_0^1(v^{-1/2}-1)dv=1$, as required.