= Singular point of a differential equation
= Singular points of a differential equation
{synonym}
For a <scalar> <linear ordinary differential equation> written in normalized form $y''+p(x)y'+q(x)y=0$, 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 $(x-a)^2q(x)$ nevertheless extend analytically across $a$. This weaker condition allows the <Frobenius method>.
For $x^2y''+xy'-y=0$, 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.
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