Frobenius & Special Functions · Superchallenge problem

The equation $xy'' + y' - y = 0$ has a double indicial root $r = 0$ at $x = 0$. Let $y(x) = \sum_{n\ge0}a_nx^n$ be the Frobenius solution with $a_0 = 1$. Compute $\displaystyle\int_0^\infty e^{-2x}\,y(x)\,dx$.