look for the set of transformations of variables which leave the ODE invariant
transinv([xi(x, y), eta(x, y)], y(x), s(r))
[xi(x, y), eta(x, y)]
list of the coefficients of the infinitesimal symmetry generator (infinitesimals)
new dependent variable
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](..).
An ODE with an arbitrary function F
ODE_y ≔ ⅆⅆx⁢y⁡x=2⁢a−x2⁢y⁡x+x2⋅2⁢F⁡x⁢y⁡x2−4⁢ax⁢a
A pair of infinitesimals for it
infinitesimals ≔ symgen⁡ODE_y
The transformation of variables which leaves ODE_y invariant
tr ≔ transinv⁡infinitesimals,y⁡x,s⁡r
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 ≔ solve⁡tr,x,y⁡x
The change of variables
ODE_s ≔ dchange⁡itr,ODE_y,r,s⁡r:
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").
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