Frobenius & Special Functions · Superchallenge problem

Bessel's equation of order $1$, $x^2y'' + xy' + (x^2 - 1)y = 0$, has indicial roots $\pm1$ at $x = 0$. Let $y_1 = x - \dfrac{x^3}{8} + \cdots$ be the Frobenius solution for $r = 1$. A second solution has the form $y_2 = C\,y_1\ln x + x^{-1}\displaystyle\sum_{n\ge0}b_nx^n$ with $b_0 = 1$. Find $C$. Express your answer as a common fraction.