 Singular point of a differential equation

For a scalar linear ordinary differential equation written in normalized form $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0$, an ordinary point has coefficients that are analytic functions $p,q$ in a neighborhood; a point where this fails is a singular point of a differential equation. It is a regular singular point if $(x-a)p(x)$ and $(x-a{)}^{2}q(x)$ nevertheless extend analytically across $a$. This weaker condition allows the Frobenius method.

For $x^2{y}^{\prime \prime }+x{y}^{\prime }-y=0$, the origin is regular singular, and the solutions $x$ and $x^{-1}$ demonstrate that an equation's singular point need not be singular for every solution. The classification concerns the normalized equation, rather than any one selected solution.