---
tag: QK2J038VB4
kind: concept
name: Ordinary point and singular point
msc: [34M03, 34A25]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_method
---

## Summary

For a linear differential equation written with leading coefficient $1$, a point is ordinary when all the coefficients are analytic there, and singular otherwise.
At an ordinary point the solutions are themselves analytic, and they can be found as [[power series|08RT9GDDPQ]].
At a singular point the solutions may blow up, branch or oscillate wildly.
The singular points are split further into [[regular|H96HXG6CA0]] and [[irregular|AT1W4MMTYR]] ones.
See the [[definition|T01779NV3Z]].

## Explanations

### L1 human

Divide the equation by the coefficient of the highest derivative.
Where the new coefficients are nice power series, nothing special happens: the point is ordinary.
Where one of them blows up, the point is singular.
For $x y'' + y = 0$, dividing by $x$ gives the coefficient $\frac{1}{x}$, so $x = 0$ is a singular point and every other point is ordinary.