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rational_series_sol

  

formal power series solutions with rational coefficients for a linear ODE

 

Calling Sequence

Parameters

Description

Options

Examples

Calling Sequence

rational_series_sol(ode, var,opts)

rational_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 rational_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 P[n]x 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 is a rational function for all sufficiently large n.

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

• 

'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.

• 

'index'=n

  

Specifies a name for the summation index in the power series. The default value is the global name _n.

Examples

withSlode:

ode12xx1ⅆⅆxⅆⅆxyx+7x3ⅆⅆxyx+2yx=0

ode1:=2xx1ⅆ2ⅆx2yx+7x3ⅆⅆxyx+2yx=0

(1)

rational_series_solode1,yx,x=0

2_C1_n=0∞_n+1x_n2_n+1

(2)

ode23xⅆⅆxⅆⅆxyxⅆⅆxyx

ode2:=3xⅆ2ⅆx2yxⅆⅆxyx

(3)

rational_series_solode2,yx,'index'=n

_C1+_C0n=1∞x2nn

(4)

An inhomogeneous equation:

ode32yx+2x+2x2ⅆ3ⅆx3yx+13x2x25ⅆ2ⅆx2yx+127xⅆⅆxyx=136x3+n=4∞xn12+13n2+4n417n3+14nn2n3n1n

ode3:=2yx+2x22xⅆ3ⅆx3yx+2x2+13x5ⅆ2ⅆx2yx+127xⅆⅆxyx=136x3+n=4∞xn4n417n3+13n2+14n12n2n3n1n

(5)

rational_series_solode3,yx,'free'=A

_n=2∞2_n3A14_n2A1+2_nA1+2_n1_n1_n2_n1x_n

(6)

See Also

LODEstruct

Slode

Slode[candidate_points]

Slode[hypergeom_series_sol]

Slode[polynomial_series_sol]

 


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