look for the set of transformations of variables which leave the ODE invariant - Maple Help

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DEtools[transinv] - look for the set of transformations of variables which leave the ODE invariant

Calling Sequence

transinv([xi(x, y), eta(x, y)], y(x), s(r))

Parameters

[xi(x, y), eta(x, y)]

-

list of the coefficients of the infinitesimal symmetry generator (infinitesimals)

y(x)

-

dependent variable

s(r)

-

new dependent variable

Description

• 

transinv looks for the set of transformations of variables which leave an ODE invariant, by using the coefficients of a symmetry generator (infinitesimals) for it. These transformations are actually the finite form of the one-parameter Lie group of invariance of the ODE.

• 

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

Examples

An ODE with an arbitrary function F

withDEtools:

withPDEtools:

ODE_y:=ⅆⅆxyx=2ax2yx+x22Fxyx24axa

ODE_y:=ⅆⅆxyx=2ax2yx+2x2Fxyx24axa

(1)

odeadvisorODE_y

_1st_order,_with_symmetry_[F(x),G(y)]

(2)

A pair of infinitesimals for it

infinitesimals:=symgenODE_y

infinitesimals:=_ξ=x2y,_η=2a

(3)

The transformation of variables which leaves ODE_y invariant

tr:=transinvinfinitesimals,yx,sr

tr:=r=x_α2ax_αxyx+1,sr=2_αa+yx

(4)

Note the introduction of _alpha, representing the parameter of the Lie group. Now, to check the invariance of ODE_y under this group, you can change variables as follows:

itr:=solvetr,x,yx

itr:=x=r1+_α2ar+sr_αr,yx=sr+2_αa

(5)

The change of variables

ODE_s:=dchangeitr,ODE_y,r,sr:

ⅆⅆrsr=solveODE_s,ⅆⅆrsr

ⅆⅆrsr=2ar22Fsr2r4arasr

(6)

As can be seen above, we arrived at the original ODE_y just changing x, y by r, s (this is the meaning of "leaving the ODE invariant").

See Also

canoni, dchange, DEtools, equinv, odeadvisor, PDEtools, symgen


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