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DEtools

  

regularsp

  

compute the regular singular points of a second order non-autonomous linear ODE

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

regularsp(des, ivar, dvar)

Parameters

des

-

second order linear ordinary differential equation or its list form

ivar

-

indicates the independent variable when des is a list with the ODE coefficients

dvar

-

indicates the dependent variable, required only when des is an ODE and the dependent variable is not obvious

Description

• 

Important: The regularsp command has been deprecated.  Use the superseding command DEtools[singularities], which computes both the regular and irregular singular points, instead.

• 

The regularsp command determines the regular singular points of a given second order linear ordinary differential equation. The ODE could be given as a standard differential equation or as a list with the ODE coefficients (see DEtools[convertAlg]). Given a linear ODE of the form

 p(x) y''(x) + q(x) y'(x) + r(x) y(x) = 0,  p(x) <> 0,  p'(x) <> 0

  

a point alpha is considered to be a regular singular point if

1) alpha is a singular point,

2) limit( (x-alpha)*q(x)/p(x), x=alpha ) = 0 and

   limit( (x-alpha)^2*r(x)/p(x), x=alpha ) = 0.

• 

The results are returned in a list.  In the event that no regular singular points are found, an empty list is returned.

Examples

Important: The regularsp command has been deprecated.  Use the superseding command DEtools[singularities], which computes both the regular and irregular singular points, instead.

withDEtools&colon;

An ordinary differential equation (ODE)

ODE&DifferentialD;2&DifferentialD;x2yx&equals;&alpha;x1&plus;&beta;x&plus;&gamma;x2&plus;&delta;x12&plus;&lambda;2yx

ODE:=&DifferentialD;2&DifferentialD;x2yx&equals;&alpha;x1&plus;&beta;x&plus;&gamma;x2&plus;&delta;x12&plus;&lambda;2yx

(1)

regularspODE

Warning, DEtools[regularsp] has been superseded by DEtools[singularities]

0&comma;1

(2)

singularitiesODE

regular&equals;0&comma;1&comma;irregular&equals;&infin;

(3)

The coefficient list form

coefs21x2x&plus;1&comma;0&comma;100x2x12&colon;

regularspcoefs&comma;x

0&comma;1&comma;&infin;

(4)

singularitiescoefs&comma;x

regular&equals;0&comma;1&comma;&infin;&comma;irregular&equals;

(5)

You can convert convert an ODE to the coefficient list form using DEtools[convertAlg] form

ODE2x2&plus;5x3&DifferentialD;2&DifferentialD;x2yx&plus;5xx2&DifferentialD;&DifferentialD;xyx&plus;1x&plus;xyx&equals;0

ODE:=5x3&plus;2x2&DifferentialD;2&DifferentialD;x2yx&plus;x2&plus;5x&DifferentialD;&DifferentialD;xyx&plus;1x&plus;xyx&equals;0

(6)

LconvertAlgODE&comma;yx

L:=1x&plus;x&comma;x2&plus;5x&comma;5x3&plus;2x2&comma;0

(7)

regularspL&comma;x

25&comma;&infin;

(8)

singularitiesL&comma;x

regular&equals;25&comma;&infin;&comma;irregular&equals;0

(9)

See Also

DEtools

DEtools[convertAlg]

DEtools[indicialeq]

DEtools[singularities]

 


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