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gensys

  

return the determining PDE system for either the coefficients of the symmetry generator or the integrating factor

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

gensys(ODE, [_xi = f(x, y, ..), _eta = g(x, y, ..)], y(x))

gensys(ODE, _mu = f(x, y, ..), y(x))

Parameters

ODE

-

ODE (ordinary differential equation)

[_xi=.., _eta=..]

-

(optional) list containing the infinitesimal; right hand sides may be functions or algebraic expressions

_mu =..]

-

(optional) equation where the right hand side may be a function or an algebraic expression

y(x)

-

(optional) required if the ODE has derivatives of more than one function

Description

• 

Given an ODE, gensys returns the determining PDE system related to either its symmetries or its integrating factors, depending on the second argument. This command is typically used together with other commands to determine symmetries and integrating factors for ODEs. gensys first calls odepde to get the determining PDE and then splits it -- when possible -- by taking coefficients with respect to the derivatives of the dependent variable.

• 

Concerning the optional second argument, gensys works like odepde; this second argument may be either an explicit form for the symmetry or the integrating factor to be taken into account at the time of calculating the determining PDE.

• 

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

Examples

withDEtools:

withPDEtools:

declareyx,prime=x

yxwill now be displayed asy

derivatives with respect toxof functions of one variable will now be displayed with '

(1)

1. Determine if an integrating factor of the form μx,y exists for the second order nonlinear ODE

ODEⅆ2ⅆx2yx=ⅇxⅆⅆxyxxyx2+yx2ⅆⅆxyxxxyx2ⅇx

ODE:=y''=ⅇxy'xy2+y2y'xxy2ⅇx

(2)

and if so, determine the integrating factor itself.

Set up the determining PDE system for these integrating factors (gensys enters in this step), then try to simplify it (perhaps using casesplit), then try to solve it. The determining PDE system is set up as follows:

gensysODE,_μ=μx,yx

ⅇxμyx,μy,yⅇxx,μx,yⅇxx,ⅇxy21μxxy2μx,yxy2+μx,xⅇxx+μy

(3)

This system (each equation above is assumed to be equal to zero) has four equations,

nops

4

(4)

and by taking into account their integrability conditions it is simplified to

casesplit

μx=μx,y,μy=0 &where

(5)

Thus, mu does not depend on y and the remaining equation leads to

μ=ⅇx

μ=ⅇx

(6)

Note that this result can be tested using mutest

mutestⅇx,ODE

0

(7)

and can be used to obtain a first integral (that is, to reduce the order of ODE).

firintⅇxODE

1y+ⅇxy'+lnx+_C1=0

(8)

First integrals can be tested using firtest.

2. Determine if an integrating factor of the form muy,y' exists for the same second order ODE.

gensysODE,_μ=μy,_y1

μ_y1,yⅇx_y1x2y3μ_y1,_y1ⅇx_y1x2y3+2μyⅇxx2y32μ_y1ⅇxx2y3+μ_y1,_y1_y1x2yμ_y1,_y1xy3+2μ_y1x2y,μ_y1,yⅇx_y12x2y3+_y12μy,yⅇxx2y3+μ_y1,y_y12x2yμ_y1,y_y1xy3+μyxy32μ_y1_y12x2μ_y1_y1x2y+μ_y1xy3μy,_y1yx2+μ_y1y3

(9)

nops

2

(10)

casesplit

μy,_y1=0 &where

(11)

So this ODE has no integrating factor of the form muy,y'.

3. Determine whether or not this ODE has point symmetries.

Set the determining system for the coefficients of the infinitesimal symmetry generator (so-called "infinitesimals") as follows:

gensysODE,ξx,y,ηx,y

ξy,yⅇxx2y3,ηy,yⅇxx2y32ξx,yⅇxx2y3+2ξyⅇxx2y32ξyx2y,2ηx,yⅇxx2y3ξx,xⅇxx2y3+ξxⅇxx2y3+3ξyxy3+ξx,yx2yξxx2y+2ηx,yx2,ηx,xⅇxx2y3+ηxⅇxx2y3ξx,yxy3+2ξxxy3ηyxy3ηxx2yξx,yy3

(12)

(in above, [_xi = xi(x,y), _eta = eta(x,y)] also works). This system consists of 4 equations,

nops

4

(13)

and in simplifying these equations by taking into account their integrability conditions one obtains

casesplit

ηx,y=0,ξx,y=0 &where

(14)

showing that this ODE has no point symmetries.

See Also

DEtools[buildsym]

DEtools[equinv]

DEtools[firint]

DEtools[firtest]

DEtools[intfactor]

DEtools[Lie]

DEtools[mutest]

DEtools[odepde]

DEtools[symgen]

dsolve[Lie]

PDEtools

PDEtools[casesplit]

pdsolve

 


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