Extension: Solving y'' + ay' + by = 0

There is a unique twice-differentiable function $g$ on the positive reals such that $g(x) + 2x^3 g'(x) + x^4 g''(x) = 0$ for all positive reals $x$, and $\lim_{x\to\infty} xg(x) = 1$. There is a unique real number $a > 1$ such that $g(a) = 1/2$. Find $\pi \cdot a$.

Source: HMMT February 2007

Hint: For intuition, write $g(x) = \sum_{n=-\infty}^{\infty} c_n x^n$ (temporarily ignoring convergence issues). Then the differential equation for $g$ gives you a linear recurrence; what are the solutions of this linear recurrence?