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singularities

  

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

 

Calling Sequence

Parameters

Description

Examples

References

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, yx, or as a list with the coefficients of yx,y'x,y''x,... (see DEtools[convertAlg]).

• 

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

A0y+A1y' +...+yn=0

  

x0 is a singular point of the equation if any of the coefficients Ai has a singularity at it. Otherwise, all the Ai are analytic at x0 and the point is an ordinary point.

• 

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

xx0niAi

  

is analytic at x=x0 for all Ai. For example, in the case of second order linear ODEs, a singularity at x0 is regular if both

xx02A0,

xx0A1

  

are analytic at x=x0.

• 

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

rⅇgular=x0,x1,,ⅈrrⅇgular=x3,x4,,FAIL=x6,x7,

  

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

• 

The nature of the point x0= is determined by changing variables x=1t: 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 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

withDEtools,singularities,hyperode,convertAlg,dpolyform

singularities,hyperode,convertAlg,dpolyform

(1)

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

ODE_2F1hyperodehypergeoma,b,c,x,yx=0

ODE_2F1:=yxab+a+b+1xcⅆⅆxyx+x2xⅆ2ⅆx2yx=0

(2)

has three regular singular points.

singularitiesODE_2F1

regular=0,1,∞,irregular=

(3)

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

ODE_1F1hyperodehypergeoma,c,x,yx=0

ODE_1F1:=ayx+c+xⅆⅆxyxxⅆ2ⅆx2yx=0

(4)

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

singularitiesODE_1F1

regular=0,irregular=∞

(5)

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

LconvertAlgODE_1F1,yx

L:=a,c+x,x,0

(6)

singularitiesL,x

regular=0,irregular=∞

(7)

Bessel equation and its singularities

Bessel_ODEop1,1,dpolyformyx=BesselJa,x,no_Fn

Bessel_ODE:=ⅆ2ⅆx2yx=ⅆⅆxyxx+a2x2yxx2

(8)

singularitiesBessel_ODE

regular=0,irregular=∞

(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 -> αx+βγx+δ in the Bessel_ODE, also has one regular and one irregular singularity:

ODE2ⅆ2ⅆx2yx=2αγx+αδ+γβⅆⅆxyxαx+βγx+δ+αδγβ2βαx+δa+γxaβ+αx+δa+γxayxγx+δ4αx+β2

ODE2:=ⅆ2ⅆx2yx=2αγx+αδ+βγⅆⅆxyxαx+βγx+δ+αδβγ2aγx+aδαxβaγx+aδ+αx+βyxγx+δ4αx+β2

(10)

singularitiesODE2

regular=βα,irregular=δγ

(11)

An example with four regular singular points

ODE37211x173+58+263yx+72xx1720x19+3x4ⅆⅆxyx+x7x4x1ⅆ2ⅆx2yx=0

ODE3:=79273x37500536792yx+727xx15336x+209ⅆⅆxyx+x7x4x1ⅆ2ⅆx2yx=0

(12)

singularitiesODE3

regular=0,1,47,∞,irregular=

(13)

References

  

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

See Also

DEtools

DEtools[convertAlg]

DEtools[dpolyform]

DEtools[hyperode]

DEtools[indicialeq]

FunctionAdvisor,singularities

singular

 


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