 Solution

For the normalized equation, $x_0$ is an ordinary point when both $p$ and $q$ are analytic near $x_0$. A singular point is a regular singular point when $(x-x_0)p(x)$ and $(x-x_0{)}^{2}q(x)$ extend analytically to $x_0$; equivalently $p$ has at most a simple pole and $q$ at most a double pole there.

Thus ordinary points admit ordinary power-series solutions, while the Frobenius method applies at a regular singular point. The definitions refer to the coefficients after dividing by the leading coefficient of $y^{\prime \prime }$.