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Slode[candidate_points] - determine points for power series solutions

Calling Sequence

candidate_points(ode, var, 'points_type'=opt)

candidate_points(lode, 'points_type'=opt)

Parameters

ode

-

linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

opt

-

(optional) type of points; one of dAlembertian, hypergeom, rational, polynomial, or all (the default).

LODEstr

-

LODEstruct data structure

Description

• 

The candidate_points command determines candidate points for which power series solutions with d'Alembertian, hypergeometric, rational, or polynomial coefficients of the given linear ordinary differential equation exist.

• 

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 either be homogeneous or have a right hand side that is rational in x

– 

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

• 

If opt=all, the output is a list of three elements:

– 

a set of hypergeometric points, which may include the symbol 'any_ordinary_point'

– 

a set of rational points;

– 

a set of polynomial points.

  

Otherwise, the output is the set of the required points.

• 

Note that the computation of candidate points for power series solutions with d'Alembertian coefficients is currently considerably more expensive computationally than for the other three types of coefficients.

Examples

withSlode:

ode:=3x26x+3ⅆⅆxⅆⅆxyx+12x12ⅆⅆxyx+6yx

ode:=3x26x+3ⅆ2ⅆx2yx+12x12ⅆⅆxyx+6yx

(1)

candidate_pointsode,yx,'type'='polynomial'

0

(2)

candidate_pointsode,yx,'type'='rational'

1

(3)

candidate_pointsode,yx,'type'='hypergeometric'

1,any_ordinary_point

(4)

candidate_pointsode,yx,'type'='all'

1,any_ordinary_point,1,0

(5)

candidate_pointsode,yx,'type'='dAlembertian'

1,any_ordinary_point

(6)

ode1:=60yx+2xx30ⅆⅆxyxx22x27ⅆ2ⅆx2yx+x34x27ⅆ3ⅆx3yx=2x25330x+60x41137x2+32x3x16

ode1:=60yx+2xx30ⅆⅆxyxx22x27ⅆ2ⅆx2yx+x34x27ⅆ3ⅆx3yx=2x260x4+32x31137x2330x5x16

(7)

Inhomogeneous equations are handled:

candidate_pointsode1,yx

0,1,274,any_ordinary_point,RootOf60_Z4+32_Z31137_Z2330_Z5,1,0,1,234,274,RootOf49_Z4287_Z31418_Z2714_Z451,RootOf49_Z4287_Z31418_Z2714_Z45,RootOf60_Z4+32_Z31137_Z2330_Z5,1,0,234,RootOf60_Z4+32_Z31137_Z2330_Z51

(8)

An equation which has d'Alembertian series solutions at any ordinary point but doesn't have hypergeometric ones:

ode2:=x1ⅆⅆxyxx2yx

ode2:=x1ⅆⅆxyxx2yx

(9)

candidate_pointsode2,yx,'type'='hypergeometric'

1

(10)

candidate_pointsode2,yx,'type'='dAlembertian'

1,any_ordinary_point

(11)

See Also

LODEstruct, Slode, Slode[candidate_mpoints]


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