Frobenius & Special Functions · Superchallenge problem

The equation $x^2y'' + 2xy' + (x^2 - 2)y = 0$ has indicial roots $1$ and $-2$ at $x = 0$. Let $y = \sum_{n\ge0}a_nx^{n+1}$ be the Frobenius solution for $r = 1$ with $a_0 = 1$. Sum the series in closed form and compute $y\!\left(\dfrac{\pi}{2}\right)$.