Consider the second-order linear ODE \begin{align*} p(x)y'' + q(x)y' + r(x)y = 0. \end{align*} 1. If $x = x_0$ is an **ordinary point** (i.e. $q/p$ and $r/p$ are analytic at $x_0$), then there exist two linearly independent solutions of the form \begin{align*} y = \sum_{n=0}^{\infty} a_n (x - x_0)^n, \end{align*} convergent in some neighbourhood of $x_0$. 2. If $x = x_0$ is a **regular singular point** (i.e. $(x - x_0)q/p$ and $(x - x_0)^2 r/p$ are analytic at $x_0$), then there exists at least one solution of the form \begin{align*} y = (x - x_0)^{\sigma} \sum_{n=0}^{\infty} a_n (x - x_0)^n \end{align*} where $\sigma \in \mathbb{R}$ and $a_0 \neq 0$. The exponent $\sigma$ is determined by the **indicial equation** obtained from the lowest-order term.