Frobenius & Special Functions · Hard problem

For $xy'' + y = 0$ the indicial roots at $x = 0$ are $r_1 = 1$ and $r_2 = 0$. Let $y_1 = x - \dfrac{x^2}{2} + \dfrac{x^3}{12} - \cdots$ be the Frobenius solution for $r_1 = 1$. A second solution has the form $y_2 = C\,y_1\ln x + \sum_{n\ge0}b_nx^n$ with $b_0 = 1$. Find $C$.