Existence of Power Set Solutions near Ordinary Points

If $x_{0}$ is an ordinary point of the Differential Equation

a_{2} (x) y^{″} + a_{1} (x) y^{'} + a_{0} (x) y = 0 ⊛

then $⊛$ has two linearly independent solutions $y_{1} (x)$ and $y_{2} (x)$ that are both analytic at $x_{0}$ :

y_{1} (x) = \sum_{n = 0}^{\infty} b_{n} (x - x_{0})^{n}, y_{2} (x) = \sum_{n = 0}^{\infty} c_{n} (x - x_{0})^{n}

and each solution has radius of convergence $R$ , where $R$ is the distance to the nearest singular point to $x_{0}$ in the complex plane.