The singularities command computes the regular and irregular singular points of a given homogeneous linear ODE. The ODE could be given as a standard differential equation in, say, , or as a list with the coefficients of  (see DEtools[convertAlg]).

Given a nth order linear homogeneous ODE with rational coefficients ,  ranging from 0 to  and ,

 is a singular point of the equation if any of the coefficients  has a singularity at it. Otherwise, all the  are analytic at  and the point is an ordinary point.

A singular point  of a nth order linear ODE can be regular or irregular. The singularity is regular whenever

is analytic at  for all . For example, in the case of second order linear ODEs, a singularity at  is regular if both

are analytic at .

The singularities command returns results as a list of equations with the singular points and their classification

The  and  equations are present in the output regardless of the sets in their right-hand sides being empty. The equation  is present only when the command failed in classifying some of the singular points.

The nature of the point  is determined by changing variables : the original ODE in  has a (regular or irregular) singularity at infinity whenever the changed ODE in  has a (regular or irregular) singularity at .

This function is part of the DEtools package, and so it can be used in the form singularities(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[singularities](..).

Ince, E.L. Ordinary Differential Equations. New York: Dover Publications, 1956.