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LinearFunctionalSystems[HomogeneousSystem] - return the homogeneous system equivalent to the given system

Calling Sequence

HomogeneousSystem(homo, sys, vars)

HomogeneousSystem(homo, A, b, x, case)

HomogeneousSystem(homo, A, x, case)

Parameters

homo

-

name; indicates whether the given system is homogeneous

sys

-

list of equations; linear functional system

vars

-

list of function variables such as [y1x,y2x,...]; variables to solve for

A

-

Matrix with rational elements

b

-

Vector with rational elements

x

-

independent variable

case

-

name indicating the case of the system; one of 'differential', 'difference', or 'qdifference'

Description

• 

The HomogeneousSystem function returns the homogeneous system that is equivalent to the given linear functional system of equations with polynomial coefficients.

  

The system parameter is entered either in list form (a list of equations sys and a list of function variables vars to solve for), or in matrix form (matrix A, vector b, and the independent variable x, where the vector b is optional).

  

The matrix form specifies the system Lyx=Ayx+b, where L is the operator (either differential, difference, or q-difference), yx is the vector of the functions to solve for, A is a rational matrix, and b is a rational vector (right-hand side).

  

For the matrix from of the calling sequence, the case of the system must be specified as one of 'differential', 'difference', or 'qdifference'. If b is not specified, the system is assumed to be homogeneous.

  

The parameter homo is set to true if the given system is homogeneous and false otherwise.

• 

All of the base functions of the LinearFunctionalSystems package are designed to handle homogeneous systems. In order to solve inhomogeneous systems, the following technique is used.

  

For a given inhomogeneous system, HomogeneousSystem constructs the equivalent homogeneous system by adding one more equation for the constant function and transforming all right-hand sides of the given system into the coefficients of this new function.

• 

The equivalent homogeneous system that is returned is in the form suitable to be used by other functions of the LinearFunctionalSystems package. Namely, it either returns a sequence of a new list of equations and a list of functions to solve for, or a new matrix, vector, variable, and case.

• 

The error conditions associated with HomogeneousSystem are the same as those which are generated by LinearFunctionalSystems[Properties].

• 

This function is part of the LinearFunctionalSystems package, and so it can be used in the form HomogeneousSystem(..) only after executing the command with(LinearFunctionalSystems). However, it can always be accessed through the long form of the command by using the form LinearFunctionalSystems[HomogeneousSystem](..).

Examples

withLinearFunctionalSystems:

sys:=ⅆⅆxy1xy2x,ⅆⅆxy2xy3xy4x,ⅆⅆxy3xy5x,ⅆⅆxy4x2y1x2xy2xy5x,ⅆⅆxy5xx2y1x2xy3xy6x,ⅆⅆxy6xx2y2x+2y3x:

vars:=y1x,y2x,y3x,y4x,y5x,y6x:

HomogeneousSystem'homo',sys,vars

ⅆⅆxy1xy2x,ⅆⅆxy2xy3xy4x,ⅆⅆxy3xy5x,ⅆⅆxy4x2y1x2xy2xy5x,ⅆⅆxy5xx2y1x2xy3xy6x,ⅆⅆxy6xx2y2x+2y3x,y1x,y2x,y3x,y4x,y5x,y6x

(1)

homo

true

(2)

sys:=ⅆⅆxy1x1+x2+6+2xy1x+6y3x6y4x=0,ⅆⅆxy2x2x1+x224+xy2x1+x22+4y3x1+x22=4x24+x1+x2,ⅆⅆxy3x1+x22x4+3x1+x2y2x+41+x2y3x=4+3x,ⅆⅆxy4x1+x2x212x1+x22x7+x22xy1xy2x4+3x1+x2x21+y3x6x+2x221+x22y4x61+x22x=4+3xx21

sys:=ⅆⅆxy1xx21+6+2xy1x+6y3x6y4x=0,2ⅆⅆxy2xxx2+124+xy2xx2+12+4y3xx2+12=4x24+xx2+1,2ⅆⅆxy3xx2+1x4+3xx2+1y2x+4x2+1y3x=43x,2ⅆⅆxy4xx2+1x21x2x2+1xx22x7y1xy2x4+3xx2+1x21+2y3x2x2+6x2x2+112y4xx2+1x=4+3xx21

(3)

vars:=y1x,y2x,y3x,y4x:

HomogeneousSystem'homo',sys,vars

ⅆⅆxy1xx2+2y1xxⅆⅆxy1x+6y3x6y4x+6y1x,y2xx5+2ⅆⅆxy2xx54y2xx4+4y3xx42y2xx3+4ⅆⅆxy2xx38x2y2x+8y3xx2xy2x+2ⅆⅆxy2xx4y2x+4y3x+x3+8x2+x+4_ycx,3y2xx3+2ⅆⅆxy3xx34x2y2x+4y3xx23xy2x+2ⅆⅆxy3xx4y2x+4y3x+4+3x_ycx,3y2xx5+2ⅆⅆxy4xx52y1xx54y2xx4+4y3xx4+4y1xx4+12y3xx312y4xx3+12y1xx3+4x2y1x+3xy2x+12xy3x12y4xx2ⅆⅆxy4xx+14y1xx+4y2x4y3x+3x3+4x23x4_ycx,ⅆⅆx_ycx,y1x,y2x,y3x,y4x,_ycx

(4)

homo

false

(5)

See Also

LinearFunctionalSystems[PolynomialSolution], LinearFunctionalSystems[Properties], LinearFunctionalSystems[RationalSolution]


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