Series Solutions · Superchallenge problem

Consider $(1 - x)^2\,y'' = 2y$ with $y(0) = 1$. For every choice of $y'(0)$ there is a power-series solution about the ordinary point $x_0 = 0$, and the theorem only guarantees convergence for $|x| < 1$. For exactly one value of $y'(0)$ the series converges for every real $x$. Find that value.