msparse_series_sol - Maple Help
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Slode

  

msparse_series_sol

  

formal m-sparse power series solutions for a linear ODE

 

Calling Sequence

Parameters

Description

Options

Examples

Calling Sequence

msparse_series_sol(ode, var, vn, opts)

msparse_series_sol(LODEstr, vn, opts)

Parameters

ode

-

linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

vn

-

new function in the form v(n)

opts

-

optional arguments of the form keyword=value

LODEstr

-

LODEstruct data structure

Description

• 

The msparse_series_sol command returns a set of m-sparse power series solutions of the given linear ordinary differential equation with polynomial coefficients.

• 

If ode is an expression, then it is equated to zero.

• 

The command returns an error message if the differential equation ode does not satisfy the following conditions.

– 

ode must be homogeneous and linear in var

– 

The coefficients of ode must be polynomial in the independent variable of var, for example, , over the rational number field which can be extended by one or more parameters.

• 

A homogeneous linear ordinary differential equation with coefficients that are polynomials in  has a linear space of formal power series solutions  where  is one of , , , or ,  is the expansion point, and the sequence  satisfies a homogeneous linear recurrence.

• 

This command selects such formal power series solutions where for an integer  there is an integer  such that

– 

 only if , and

– 

the sequence  satisfies a linear recurrence  for all sufficiently large .

• 

The m-sparse power series is represented by an FPSstruct data-structure (see Slode[FPseries]):

  

where

– 

,..., are expressions, the initial series coefficients,

– 

 is a nonnegative integer, and

– 

 is an integer such that .

Options

• 

x=a or 'point'=a

  

Specifies the expansion point a. It can be an algebraic number, depending rationally on some parameters, or .

  

If this option is given, then the command returns a set of m-sparse power series solutions at the given point a. Otherwise, it returns a set of m-sparse power series solutions for all possible points that are determined by Slode[candidate_mpoints](ode,var).

• 

'sparseorder'=m0

  

Specifies an integer m0. If this option is given, then the command computes a set of m-sparse power series solutions with  only. Otherwise, it returns a set of m-sparse power series solution for all possible values of .

  

If both an expansion point and a sparse order are given, then the command can also compute a set of m-sparse series solutions for an inhomogeneous equation with polynomial coefficients and a right-hand side that is rational in the independent variable . Otherwise, the equation has to be homogeneous.

• 

'free'=C

  

Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation.

Examples

(1)

(2)

Inhomogeneous equations are handled:

(3)

(4)

See Also

LODEstruct

Slode

Slode[candidate_mpoints]

Slode[FPseries]

 


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