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dAlembertian_series_sol

  

formal power series solutions with d'Alembertian coefficients for a linear ODE

 

Calling Sequence

Parameters

Description

Options

Examples

Compatibility

Calling Sequence

dAlembertian_series_sol(ode,var,opts)

dAlembertian_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 dAlembertian_series_sol command returns one formal power series solution or a set of formal power series solutions with d'Alembertian coefficients for 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 linear in var

– 

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 polynomial in the independent variable of var, for example, x, 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 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 command selects such formal power series solutions where vn is a d'Alembertian sequence, that is, vn is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).

• 

The command determines an integer N0 such that vn can be represented in the form of a d'Alembertian term:

vn=h1nn1=Nn1h2n1n2=Nn11...ns=Nns11hs+1ns ( + )

  

for all nN, where hin, 1is+1, is a hypergeometric term (see SumTools[Hypergeometric]):

hin=hiNk=Nn1Rk ( ++ )

  

such that Rk=hik+1hik is rational in k for all kN.

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. 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 the point is given, then the command returns one formal power series solution at a with d'Alembertian coefficients if it exists; otherwise, it returns NULL. If a  is not given, it returns a set of formal power series solutions with d'Alembertian coefficients for all singular points of ode as well as one generic ordinary point.

• 

'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.

• 

'indices'=[n,k]

  

Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).

• 

'outputHGT'=name

  

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

– 

'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).

• 

'outputDAT'=name

  

Specifies the form of representation of the sums in ( + ). The default is 'inert'.

– 

'inert' - the sums are in the inert form, except for trivial sums of the form k=uv11, which are simplified to vu.

– 

'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

Examples

withSlode:

odex2+x2diffyx,x,x+x2xdiffyx,x6x2+7xyx

odex2+x2ⅆ2ⅆx2yx+x2xⅆⅆxyx6x2+7xyx

(1)

dAlembertian_series_solode,yx,outputHGT=active,indices=n,k

_C0n=12nx1nΓn2,315n=72nn6x+2nΓn+18409541575x4315x+22315x+232105x+24221x+252_C0+17143n=72nn6n1=7n116n1Γn1+1n15n16x+2nΓn+140n=72nn6n1=7n116n1Γn1+1n2=7n11n25−9n2Γn2n15n16x+2nΓn+19+997295486+383575x486+153430x+22243+77435x+23243+73295x+24729+18835x+257292540x+26729_C1,_n=9_n82_nx+3_n512_n!252567x2563x+321285x+33384x+34192x+35640x+362880x+3720160_C0+528299_n=91024_n82_n_n1=9_n118_n1Γ_n1+1_n17_n18x+3_n2835_n!2198016+_n=92_n82_n_n1=9_n118_n1Γ_n1+1_n2=9_n11_n27−12_n2I112_n27I11+2_n2+7_n2+1Γ_n2+1_n17_n18x+3_n19683I1125I11+25_n!+869574607273468358656+243522833x273468358656+3853999x+325064228864+366367x+33859963392+1269989x+347596343296+11983373x+35227890298880+1166693x+36113945149440+572503x+3726587201536093781x+38398808023040_C1

(2)

ode2diffyx,xyx=SumSumΓ2n1+1,n1=0..n1xnn!,n=0..

ode2ⅆⅆxyxyx=n=0n1=0n1Γ2n1+1xnn!

(3)

dAlembertian_series_solode2,yx,outputHGT=active,indices=n,k

n=1n1+n1=1n1n2=1n11Γ2n2+1xnn!+_C0n=0xnn!

(4)

Compatibility

• 

The Slode[dAlembertian_series_sol] command was updated in Maple 2017.

• 

The ode parameter was updated in Maple 2017.

See Also

LinearOperators

LODEstruct

Slode

Slode[hypergeom_series_sol]

Slode[polynomial_series_sol]

Slode[rational_series_sol]

SumTools[Hypergeometric]