Frobenius & Special Functions · Hard problem

The equation $xy'' + y' + y = 0$ has a double indicial root $r = 0$ at $x = 0$, with Frobenius solution $y_1 = \displaystyle\sum_{n\ge0}\frac{(-1)^n}{(n!)^2}x^n$. A second solution has the form $y_2 = y_1\ln x + \displaystyle\sum_{n\ge1}b_nx^n$. Find $b_2$. Express your answer as a common fraction.