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matrixDE

  

find solutions of a linear system of ODEs in matrix form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

matrixDE(A, B, t, method=matrixexp)

matrixDE(A, B, t, solution=solntype)

Parameters

A, B

-

coefficients of a system X' t=AtXt+Bt ; if B not specified, then assumed to be a zero vector

t

-

independent variable of the system

method=matrixexp

-

(optional) matrix exponentials

solution=solntype

-

(optional) where solution=polynomial or solution=rational

Description

• 

The matrixDE command solves a system of linear ODEs of the form X't=AtXt+Bt. If B is not specified then it is assumed to be the zero vector.

• 

An option of the form method = matrixexp can be specified to use matrix exponentials (in the case of constant coefficients).

• 

An option of the form solution = polynomial or solution = rational can be specified to search for polynomial or rational solution. In this case, the function invokes LinearFunctionalSystems[PolynomialSolution] or LinearFunctionalSystems[RationalSolution].

  

The command returns a pair St,Pt with St, which is an n by n Matrix, and Pt, which is an n by 1 Vector. A particular solution of the system can be then written in the form Ft=StC0+Pt where C0 is n by 1 and F0=C0+P0. If B is zero then P will also be zero.

• 

If a system is expressed in terms of equations, dsolve can be used instead.

Examples

withDEtools:

Nonconstant homogeneous system

AMatrix2,2,1,t2,t,1

A1t2t1

(1)

solmatrixDEA,t

solⅇtt3/2BesselI35,25t5/2ⅇtt3/2BesselK35,25t5/2ⅇtBesselI25,25t5/2tⅇtBesselK25,25t5/2t,00

(2)

Matrix of arbitrary coefficients

CMatrix2,1:

Verification of solution

Fevalmsol1 &* C+sol2:rhevalmA &* F:

simplifynormaltF1,1rh1,1,symbolic

0

(3)

simplifynormaltF2,1rh2,1,symbolic

0

(4)

Nonhomogeneous system of two variables with constant coefficients

AMatrix2,2,1,1,0,1;BMatrix2,1,tk,tl

A1101

Btktl

(5)

solmatrixDEA,B,t

solⅇtⅇtt0ⅇt,ⅇ12tt12lWhittakerM12l,12l+12,tkl+t12l+1ⅇ12tWhittakerM12l,12l+12,tk+ⅇ12tt12kWhittakerM12k,12k+12,tlⅇ12tt12lWhittakerM12l,12l+12,tkⅇ12tt12lWhittakerM12l,12l+12,tl+t12l+1ⅇ12tWhittakerM12l,12l+12,t+ⅇ12tt12kWhittakerM12k,12k+12,tⅇ12tt12lWhittakerM12l,12l+12,t+tl+1kl+tl+1k+tl+1l+tl+1l+1k+1ⅇ12tt12lWhittakerM12l+1,12l+12,tl2+t12l+1ⅇ12tWhittakerM12l+1,12l+12,tl+ⅇ12tt12kWhittakerM12k+1,12k+12,tl2ⅇ12tt12lWhittakerM12l+1,12l+12,tl+t12l+1ⅇ12tWhittakerM12l,12l+12,t+t12l+1ⅇ12tWhittakerM12l+1,12l+12,t+ⅇ12tt12kWhittakerM12k+1,12k+12,tⅇ12tt12lWhittakerM12l+1,12l+12,ttklt+tl+1l2tl+1lttkt+2tl+1ltl+1t+tl+1l+1t

(6)

Verification of solution

Fevalmsol1 &* C+sol2:rhevalmA &* F+B:

simplifynormaltF1,1rh1,1,symbolic

0

(7)

simplifynormaltF2,1rh2,1,symbolic

0

(8)

Nonconstant homogeneous system with unknown coefficients

AMatrix2,2,1,0,1,ft

A101ft

(9)

solmatrixDEA,t

solftDESolⅆ2ⅆt2_Ytⅆⅆt_Ytftⅆⅆt_Yt_Ytⅆⅆtft+ft_Yt,_Yt+ⅆⅆtDESolⅆ2ⅆt2_Ytⅆⅆt_Ytftⅆⅆt_Yt_Ytⅆⅆtft+ft_Yt,_YtDESolⅆ2ⅆt2_Ytⅆⅆt_Ytftⅆⅆt_Yt_Ytⅆⅆtft+ft_Yt,_Yt,00

(10)

General nonhomogeneous system of two variables with constant coefficients

AMatrix2,2,a,b,c,d

Aabcd

(11)

BMatrix2,1,ft,gt

Bftgt

(12)

solmatrixDEA,B,t

solⅇ12ad+a22ad+4bc+d2tⅇ12a+d+a22ad+4bc+d2t12ⅇ12ad+a22ad+4bc+d2td+a+a22ad+4bc+d2b12ⅇ12a+d+a22ad+4bc+d2tda+a22ad+4bc+d2b,∫ftd+ⅆⅆtft+bgtⅇ12a+d+a22ad+4bc+d2tⅆtⅇta22ad+4bc+d2∫ftd+ⅆⅆtft+bgtⅇ12ad+a22ad+4bc+d2tⅆtⅇ12ad+a22ad+4bc+d2ta22ad+4bc+d2122ⅇ12a+d+a22ad+4bc+d2tftⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d2d2ⅇ12a+d+a22ad+4bc+d2tⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d2gtba22ad+4bc+d2ⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d2∫ftd+ⅆⅆtft+bgtⅇ12a+d+a22ad+4bc+d2tⅆt2ⅇ12a+d+a22ad+4bc+d2tⅆⅆtftⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d22ftⅇ12ad+a22ad+4bc+d2tⅇ12ad+a22ad+4bc+d2td+ⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d2∫ftd+ⅆⅆtft+bgtⅇ12a+d+a22ad+4bc+d2tⅆtaⅇ12ad+a22ad+4bc+d2tⅇta22ad+4bc+d2∫ftd+ⅆⅆtft+bgtⅇ12a+d+a22ad+4bc+d2tⅆtd+2ⅇ12ad+a22ad+4bc+d2tⅇ12ad+a22ad+4bc+d2tgtba22ad+4bc+d2ⅇ12ad+a22ad+4bc+d2t∫ftd+ⅆⅆtft+bgtⅇ12ad+a22ad+4bc+d2tⅆt+2ⅆⅆtftⅇ12ad+