Frobenius & Special Functions · Superchallenge problem

The equation $x^2y'' + xy' + (1 + x)y = 0$ has complex indicial roots $\pm i$ at $x = 0$. Let $w = \displaystyle\sum_{n\ge0}a_nx^{n+i}$ be the Frobenius solution for $r = i$ with $a_0 = 1$ (for $x > 0$, $x^{n+i} = x^ne^{i\ln x}$), and let $y = \operatorname{Re}w$, a real solution. Writing $y = \cos(\ln x)\left(1 + p_1x + p_2x^2 + \cdots\right) + \sin(\ln x)\left(q_1x + q_2x^2 + \cdots\right)$, find $q_2$. Express your answer as a common fraction.