formal power series solutions with hypergeometric coefficients for a linear ODE - Maple Help

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Slode[hypergeom_series_sol] - formal power series solutions with hypergeometric coefficients for a linear ODE

Calling Sequence

hypergeom_series_sol(ode, var,opts)

hypergeom_series_sol(LODEstr,opts)

Parameters

ode

-

linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

opts

-

optional arguments of the form keyword=value

LODEstr

-

LODEstruct data structure

Description

• 

The hypergeom_series_sol command returns one formal power series solution or a set of formal power series solutions of the given linear ordinary differential equation with polynomial coefficients. The ODE must be either homogeneous or inhomogeneous with a right-hand side that is a polynomial, a rational function, or a "nice" power series (see LODEstruct) in the independent variable x.

• 

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

• 

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

– 

ode must be linear in var

– 

ode must have polynomial coefficients in x

– 

ode must be homogeneous or have a right-hand side that is rational or a "nice" power series in x

– 

The coefficients of ode must be either rational numbers or depend rationally on one or more parameters.

• 

A homogeneous linear ordinary differential equation with coefficients that are polynomials in x has a linear space of formal power series solutions n=0vnPnx where Pnx is one of xan, xann!, 1xn, or 1xnn!, a is the expansion point, and the sequence vn satisfies a homogeneous linear recurrence. In the case of an inhomogeneous equation with a right-hand side that is a "nice" power series, vn satisfies an inhomogeneous linear recurrence.

• 

The routine selects such formal power series solutions where vn+1=pnvn for all sufficiently large n, where pn is a rational function.

• 

This routine determines an integer N0 such that vn can be represented in the form of hypergeometric term (see SumTools[Hypergeometric],LREtools):

vn=vNk=Nn1pk ( * )

  

for all nN.

Options

• 

x=a or 'point'=a

  

Specifies the expansion point in the case of a homogeneous equation or an inhomogeneous equation with rational right-hand side. The default is a=0. It can be an algebraic number, depending rationally on some parameters, or . In the case of a "nice" series right-hand side the expansion point is given by the right-hand side and cannot be changed.

  

If this option is given, then the command returns one formal power series solution at a with hypergeometric coefficients if it exists; otherwise, it returns NULL. If a is not given, it returns a set of formal power series solutions with hypergeometric coefficients for all possible points that are determined by Slode[candidate_points](ode,var,'type'='hypergeometric').

• 

'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 if the expansion point is singular.

• 

'indices'=[n,k]

  

Specifies names for dummy variables. The default values are the global names _n and _k. The name n is used as the summation index in the power series. The name k is used as the product index in ( * ).

• 

'outputHGT'=name

  

Specifies the form of representation of hypergeometric terms.  The default value is 'active'.

– 

'inert' - the hypergeometric term ( * ) is represented by an inert product, except for k=Nn11, which is simplified to 1.

– 

'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).

– 

'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).

Examples

withSlode:

ode:=2xx1ⅆⅆxⅆⅆxyx+7x3ⅆⅆxyx+2yx=0

ode:=2xx1ⅆ2ⅆx2yx+7x3ⅆⅆxyx+2yx=0

(1)

hypergeom_series_solode,yx,x=1

_C0_n=0∞Γ12+_nx+1_n_n!π

(2)

hypergeom_series_solode,yx,x=0

_C0_n=0∞_n+1x_n2_n+1

(3)

hypergeom_series_solode,yx

_C0_n=0∞_n+1x_n2_n+1,_C0_n=0∞Γ12+_nx+1_nπ_n!,_C0_n=0∞Γ12+_n1_nx1_nΓ_n+1π

(4)

Inhomogeneous equations are handled:

ode1:=81x343x2ⅆ3ⅆx3yx+567x2439x2ⅆ2ⅆx2yx+207x452ⅆⅆxyx+45yx=35x4+330x3+1137x232x602x16

ode1:=814x33x2ⅆ3ⅆx3yx+5674x2392xⅆ2ⅆx2yx+207x452ⅆⅆxyx+45yx=325x4+330x3+1137x232x60x16

(5)

hypergeom_series_solode1,yx,x=0,'indices'=n,k

54_C03n=0∞Γn+43Γn+5327nxnΓ2n+5π94n=0∞4n+7xnn+1n+2+14n=0∞2n+132n+13xnn+1n+2+758n=0∞k=0n14k3+12k2+12k+29k+1k+34k3+25xn+116n=0∞I142nI14+2n2n+142n+14xnn+1n+2

(6)

See Also

LODEstruct, Slode, Slode[candidate_points], Slode[mhypergeom_series_sol], Slode[polynomial_series_sol], Slode[rational_series_sol]


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