Exact Equations & Substitutions · Superchallenge problem

The equation $(xy^2 - 1)\,dx + 2x^2y\,dy = 0$ has an integrating factor depending on $x$ alone. Let $y(x)$ be its solution with $y(1) = 2$, for $x > 0$ and $y > 0$. Find the maximum value of $y(x)$.