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candidate_mpoints

  

determine m-points for m-sparse power series solutions

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

candidate_mpoints(ode, var)

candidate_mpoints(LODEstr)

Parameters

ode

-

homogeneous linear ODE with polynomial coefficients

var

-

dependent variable, for example y(x)

LODEstr

-

LODEstruct data structure

Description

• 

The candidate_mpoints command determines for all positive integers m candidate points for m-sparse power series solutions of the given homogeneous linear ordinary differential equation with polynomial coefficients, called m-points.

• 

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 homogeneous and linear in var

– 

ode must have polynomial coefficients in the independent variable of var, for example, x

– 

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

• 

This command returns a list of lists with three elements:

– 

an integer mi>1, the sparse order;

– 

a LODEstruct representing an mi-sparse differential equation with constant coefficients which is a right factor of the given equation;

– 

a set of candidate mi-points.

  

The list is sorted by sparse order.

• 

If for some sparse-order m the given equation has a nontrivial m-sparse right factor with constant coefficients, then the equation has m-sparse power series solutions at an arbitrary point, and these solutions are solutions of this right factor. If the set of candidate m-points is not empty, then the equation may or may not have m-sparse power series solutions at such a point, but it does not have m-sparse power series solutions at any point outside this set.

Examples

withSlode:

ode2+x2ⅆ3ⅆx3yx2ⅆ2ⅆx2yxx+2+x2ⅆⅆxyx2xyx

ode:=x2+2ⅆ3ⅆx3yx2ⅆ2ⅆx2yxx+x2+2ⅆⅆxyx2xyx

(1)

candidate_mpointsode,yx

2,LODEstructyx+ⅆ2ⅆx2yx,yx,0

(2)

See Also

LODEstruct

Slode

Slode[candidate_points]

Slode[msparse_series_sol]

 


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