Exact Equations & Substitutions · Superchallenge problem

The equation $y^2\,dx - x\left(x + y^2 + 2y\right)dy = 0$ has no integrating factor depending on $x$ alone or on $y$ alone, but it has one of the form $\mu = f(x)\,g(y)$. Its solution through $(1, 1)$ is defined for all $x > 0$, increases, and has a horizontal asymptote $y = L$ as $x \to \infty$. Find $\displaystyle\lim_{x\to\infty} x\,\bigl(L - y(x)\bigr)$.