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, $y\left(x\right)$, or as a list with the coefficients of $y\left(x\right),y\'\left(x\right),y\text{'}\text{'}\left(x\right),\mathrm{...}$ (see DEtools[convertAlg]).

Given a nth order linear homogeneous ODE with rational coefficients $A_i$, $i$ ranging from 0 to $n$ and $A_n=1$,

$\mathrm{x0}$ is a singular point of the equation if any of the coefficients $A_i$ has a singularity at it. Otherwise, all the $A_i$ are analytic at $\mathrm{x0}$ and the point is an ordinary point.

A singular point $\mathrm{x0}$ of a nth order linear ODE can be regular or irregular. The singularity is regular whenever

is analytic at $x=\mathrm{x0}$ for all $A_i$. For example, in the case of second order linear ODEs, a singularity at $\mathrm{x0}$ is regular if both

are analytic at $x=\mathrm{x0}$.

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

The $\mathrm{regular}=\{...\}$ and $\mathrm{irregular}=\{...\}$ equations are present in the output regardless of the sets in their right-hand sides being empty. The equation $\mathrm{FAIL}=\{...\}$ is present only when the command failed in classifying some of the singular points.

The nature of the point $\mathrm{x0}=\mathrm{\infty }$ is determined by changing variables $x=\frac{1}{t}$: the original ODE in $x$ has a (regular or irregular) singularity at infinity whenever the changed ODE in $t$ has a (regular or irregular) singularity at $\mathrm{t0}=0$.

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](..).

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{singularities}\,\mathrm{hyperode}\,\mathrm{convertAlg}\,\mathrm{dpolyform}\right)$

$\left[\mathrm{singularities}\,\mathrm{hyperode}\,\mathrm{convertAlg}\,\mathrm{dpolyform}\right]$

$\mathrm{ODE\_2F1}≔\mathrm{hyperode}\left(\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)\,y\left(x\right)\right)=0$

$\mathrm{ODE\_2F1}≔y\left(x\right)ab+\left(\left(a+b+1\right)x-c\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^2-x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)=0$

$\mathrm{singularities}\left(\mathrm{ODE\_2F1}\right)$

$\mathrm{regular}=\left\{0\,1\,\mathrm{\infty }\right\},\mathrm{irregular}=\varnothing$

$\mathrm{ODE\_1F1}≔\mathrm{hyperode}\left(\mathrm{hypergeom}\left(\left[a\right]\,\left[c\right]\,x\right)\,y\left(x\right)\right)=0$

$\mathrm{ODE\_1F1}≔ay\left(x\right)+\left(-c+x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)=0$

$\mathrm{singularities}\left(\mathrm{ODE\_1F1}\right)$

$\mathrm{regular}=\left\{0\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

$L≔\mathrm{convertAlg}\left(\mathrm{ODE\_1F1}\,y\left(x\right)\right)$

$L≔\left[\left[a\,-c+x\,-x\right]\,0\right]$

$\mathrm{singularities}\left(L\,x\right)$

$\mathrm{regular}=\left\{0\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

$\mathrm{Bessel\_ODE}≔\mathrm{op}\left(\left[1\,1\right]\,\mathrm{dpolyform}\left(y\left(x\right)=\mathrm{BesselJ}\left(a\,x\right)\,\mathrm{no\_Fn}\right)\right)$

$\mathrm{Bessel\_ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=-\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{x}+\frac{\left(a^2-x^2\right)y\left(x\right)}{x^2}$

$\mathrm{singularities}\left(\mathrm{Bessel\_ODE}\right)$

$\mathrm{regular}=\left\{0\right\},\mathrm{irregular}=\left\{\mathrm{\infty }\right\}$

$\mathrm{ODE2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=-\frac{2\mathrm{\alpha }\mathrm{\gamma }x+\mathrm{\alpha }\mathrm{\delta }+\mathrm{\gamma }\mathrm{\beta }}{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}\mathrm{diff}\left(y\left(x\right)\,x\right)+\frac{{\left(\mathrm{\alpha }\mathrm{\delta }-\mathrm{\gamma }\mathrm{\beta }\right)}^2\left(-\mathrm{\beta }-\mathrm{\alpha }x+\mathrm{\delta }a+\mathrm{\gamma }xa\right)\left(\mathrm{\beta }+\mathrm{\alpha }x+\mathrm{\delta }a+\mathrm{\gamma }xa\right)}{{\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}^4{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)}^2}y\left(x\right)$

$\mathrm{ODE2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=-\frac{\left(2\mathrm{\alpha }\mathrm{\gamma }x+\mathrm{\alpha }\mathrm{\delta }+\mathrm{\gamma }\mathrm{\beta }\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}+\frac{{\left(\mathrm{\alpha }\mathrm{\delta }-\mathrm{\gamma }\mathrm{\beta }\right)}^2\left(\mathrm{\gamma }xa+\mathrm{\delta }a-\mathrm{\alpha }x-\mathrm{\beta }\right)\left(\mathrm{\gamma }xa+\mathrm{\delta }a+\mathrm{\alpha }x+\mathrm{\beta }\right)y\left(x\right)}{{\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}^4{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)}^2}$

$\mathrm{singularities}\left(\mathrm{ODE2}\right)$

$\mathrm{regular}=\left\{-\frac{\mathrm{\beta }}{\mathrm{\alpha }}\right\},\mathrm{irregular}=\left\{-\frac{\mathrm{\delta }}{\mathrm{\gamma }}\right\}$

$\mathrm{ODE3}≔-\left(\frac{72\cdot 11}{73}\left(x-1\right)+\frac{5}{8}+\frac{2}{63}\right)y\left(x\right)+\left(\frac{72}{7}x\left(x-1\right)-\frac{20}{9}\left(x-1\right)+\frac{3x}{4}\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+x\left(7x-4\right)\left(x-1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=0$

$\mathrm{ODE3}≔-\left(\frac{792x}{73}-\frac{375005}{36792}\right)y\left(x\right)+\left(\frac{72x\left(x-1\right)}{7}-\frac{53x}{36}+\frac{20}{9}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+x\left(7x-4\right)\left(x-1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)=0$

$\mathrm{singularities}\left(\mathrm{ODE3}\right)$

$\mathrm{regular}=\left\{0\,1\,\frac{4}{7}\,\mathrm{\infty }\right\},\mathrm{irregular}=\varnothing$

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