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DEtools

 singularities
 compute the regular and irregular singular points of a homogeneous linear ordinary differential equation (ODE)

 Calling Sequence singularities(ODE, y(x)) singularities(L, x)

Parameters

 ODE - linear homogeneous ordinary differential equation y(x) - optional, the dependent variable, required when the ODE contains derivatives of more than one unknown function L - list with the coefficients of y, y', ... entering the ODE x - independent variable, required only when there is more than one symbol entering the list with the ODE coefficients

Description

 • 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 ${A}_{n}=1$,

${A}_{0}y+{A}_{1}y\text{'}+\mathrm{...}+{y}^{\left(n\right)}=0$

 $\mathrm{x0}$ 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

${\left(x-\mathrm{x0}\right)}^{n-i}{A}_{i}$

 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

${\left(x-\mathrm{x0}\right)}^{2}{A}_{0},$

$\left(x-\mathrm{x0}\right){A}_{1}$

 are analytic at $x=\mathrm{x0}$.
 • The singularities command returns results as a list of equations with the singular points and their classification

$rⅇgular=\left\{x0,x1,\dots \right\},irrⅇgular=\left\{x3,x4,\dots \right\},FAIL=\left\{x6,x7,\dots \right\}$

 The $\mathrm{regular}=\left\{...\right\}$ and $\mathrm{irregular}=\left\{...\right\}$ equations are present in the output regardless of the sets in their right-hand sides being empty. The equation $\mathrm{FAIL}=\left\{...\right\}$ 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{singularities},\mathrm{hyperode},\mathrm{convertAlg},\mathrm{dpolyform}\right)$
 $\left[{\mathrm{singularities}}{,}{\mathrm{hyperode}}{,}{\mathrm{convertAlg}}{,}{\mathrm{dpolyform}}\right]$ (1)

The 2F1 hypergeometric equation, that is, the linear ODE whose solutions involve special functions of the 2F1 class

 > $\mathrm{ODE_2F1}≔\mathrm{hyperode}\left(\mathrm{hypergeom}\left(\left[a,b\right],\left[c\right],x\right),y\left(x\right)\right)=0$
 ${\mathrm{ODE_2F1}}{≔}{y}{}\left({x}\right){}{a}{}{b}{+}\left(\left({a}{+}{b}{+}{1}\right){}{x}{-}{c}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({{x}}^{{2}}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (2)

has three regular singular points.

 > $\mathrm{singularities}\left(\mathrm{ODE_2F1}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{\mathrm{\infty }}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (3)

The equation that has special functions of the 1F1 class as a solution, that is, the confluent 1F1 hypergeometric equation

 > $\mathrm{ODE_1F1}≔\mathrm{hyperode}\left(\mathrm{hypergeom}\left(\left[a\right],\left[c\right],x\right),y\left(x\right)\right)=0$
 ${\mathrm{ODE_1F1}}{≔}{a}{}{y}{}\left({x}\right){+}\left({-}{c}{+}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (4)

has one regular singularity at zero and one irregular at infinity.

 > $\mathrm{singularities}\left(\mathrm{ODE_1F1}\right)$
 ${\mathrm{regular}}{=}\left\{{0}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (5)

You can input the ODE as a list of coefficients of the unknown of the equation and its derivatives (see DEtools[convertAlg])

 > $L≔\mathrm{convertAlg}\left(\mathrm{ODE_1F1},y\left(x\right)\right)$
 ${L}{≔}\left[\left[{a}{,}{-}{c}{+}{x}{,}{-}{x}\right]{,}{0}\right]$ (6)
 > $\mathrm{singularities}\left(L,x\right)$
 ${\mathrm{regular}}{=}\left\{{0}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (7)

Bessel equation and its singularities

 > $\mathrm{Bessel_ODE}≔\mathrm{op}\left(\left[1,1\right],\mathrm{dpolyform}\left(y\left(x\right)=\mathrm{BesselJ}\left(a,x\right),\mathrm{no_Fn}\right)\right)$
 ${\mathrm{Bessel_ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)}{{x}}{+}\frac{\left({{a}}^{{2}}{-}{{x}}^{{2}}\right){}{y}{}\left({x}\right)}{{{x}}^{{2}}}$ (8)
 > $\mathrm{singularities}\left(\mathrm{Bessel_ODE}\right)$
 ${\mathrm{regular}}{=}\left\{{0}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (9)

Fractional linear transformations, also called Mobius transformations, do not change the structure of the singularities, they only move the locations of the poles. So, this other equation, obtained by changing variables $x$ -> $\frac{\mathrm{\alpha }x+\mathrm{\beta }}{\mathrm{\gamma }x+\mathrm{\delta }}$ in the Bessel_ODE, also has one regular and one irregular singularity:

 > $\mathrm{ODE2}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=-\frac{2\mathrm{\alpha }\mathrm{\gamma }x+\mathrm{\alpha }\mathrm{\delta }+\mathrm{\gamma }\mathrm{\beta }}{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}\mathrm{diff}\left(y\left(x\right),x\right)+\frac{{\left(\mathrm{\alpha }\mathrm{\delta }-\mathrm{\gamma }\mathrm{\beta }\right)}^{2}\left(-\mathrm{\beta }-\mathrm{\alpha }x+\mathrm{\delta }a+\mathrm{\gamma }xa\right)\left(\mathrm{\beta }+\mathrm{\alpha }x+\mathrm{\delta }a+\mathrm{\gamma }xa\right)}{{\left(\mathrm{\gamma }x+\mathrm{\delta }\right)}^{4}{\left(\mathrm{\alpha }x+\mathrm{\beta }\right)}^{2}}y\left(x\right)$
 ${\mathrm{ODE2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\left({2}{}{\mathrm{\alpha }}{}{\mathrm{\gamma }}{}{x}{+}{\mathrm{\alpha }}{}{\mathrm{\delta }}{+}{\mathrm{\gamma }}{}{\mathrm{\beta }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{\left({\mathrm{\alpha }}{}{x}{+}{\mathrm{\beta }}\right){}\left({\mathrm{\gamma }}{}{x}{+}{\mathrm{\delta }}\right)}{+}\frac{{\left({\mathrm{\alpha }}{}{\mathrm{\delta }}{-}{\mathrm{\gamma }}{}{\mathrm{\beta }}\right)}^{{2}}{}\left({\mathrm{\gamma }}{}{x}{}{a}{+}{\mathrm{\delta }}{}{a}{-}{\mathrm{\alpha }}{}{x}{-}{\mathrm{\beta }}\right){}\left({\mathrm{\gamma }}{}{x}{}{a}{+}{\mathrm{\delta }}{}{a}{+}{\mathrm{\alpha }}{}{x}{+}{\mathrm{\beta }}\right){}{y}{}\left({x}\right)}{{\left({\mathrm{\gamma }}{}{x}{+}{\mathrm{\delta }}\right)}^{{4}}{}{\left({\mathrm{\alpha }}{}{x}{+}{\mathrm{\beta }}\right)}^{{2}}}$ (10)
 > $\mathrm{singularities}\left(\mathrm{ODE2}\right)$
 ${\mathrm{regular}}{=}\left\{{-}\frac{{\mathrm{\beta }}}{{\mathrm{\alpha }}}\right\}{,}{\mathrm{irregular}}{=}\left\{{-}\frac{{\mathrm{\delta }}}{{\mathrm{\gamma }}}\right\}$ (11)

An example with four regular singular points

 > $\mathrm{ODE3}≔-\left(\frac{72\cdot 11}{73}\left(x-1\right)+\frac{5}{8}+\frac{2}{63}\right)y\left(x\right)+\left(\frac{72}{7}x\left(x-1\right)-\frac{20}{9}\left(x-1\right)+\frac{3x}{4}\right)\mathrm{diff}\left(y\left(x\right),x\right)+x\left(7x-4\right)\left(x-1\right)\mathrm{diff}\left(y\left(x\right),x,x\right)=0$
 ${\mathrm{ODE3}}{≔}{-}\left(\frac{{792}{}{x}}{{73}}{-}\frac{{375005}}{{36792}}\right){}{y}{}\left({x}\right){+}\left(\frac{{72}{}{x}{}\left({x}{-}{1}\right)}{{7}}{-}\frac{{53}{}{x}}{{36}}{+}\frac{{20}}{{9}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{x}{}\left({7}{}{x}{-}{4}\right){}\left({x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (12)
 > $\mathrm{singularities}\left(\mathrm{ODE3}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}\frac{{4}}{{7}}{,}{\mathrm{\infty }}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (13)

References

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