= Solution
For the normalized second-order equation, $x_0$ is an <ordinary point> if both coefficient functions $p$ and $q$ are <holomorphic> near $x_0$. If it is singular but $(x-x_0)p(x)$ and $(x-x_0)^2q(x)$ are <holomorphic> there, it is a <regular singular point>. Thus $p$ may have at most a simple pole and $q$ at most a double pole.
The distinction tells us which local series method is justified. At an <ordinary point>, two independent solutions are represented by ordinary <Taylor series>. At a <regular singular point>, the <Frobenius method> uses $(x-x_0)^r\sum_{n\geq0}a_n(x-x_0)^n$; the <indicial equation> determines the exponents, and logarithmic terms may appear in a second solution. Singularities worse than these are irregular and need not have this Frobenius form.
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