Extension: Guessing Taylor Series Solutions to Differential Equations

(HMMT 2007 Calculus #9, slightly modified) $g$ is a twice differentiable function over the positive reals such that: $$g(x) + 2x^3 g'(x) + x^4 g''(x) = 0 \text{ for all positive reals $x$.}$$ $$\lim_{x\rightarrow\infty} xg(x) = 1.$$ Solve for $g(x).$