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DEtools[transinv] - look for the set of transformations of variables which leave the ODE invariant
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Calling Sequence
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transinv([xi(x, y), eta(x, y)], y(x), s(r))
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Parameters
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[xi(x, y), eta(x, y)]
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list of the coefficients of the infinitesimal symmetry generator (infinitesimals)
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y(x)
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dependent variable
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s(r)
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new dependent variable
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Description
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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.
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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](..).
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Examples
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An ODE with an arbitrary function F
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![[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]](/support/helpjp/helpview.aspx?si=6703/file01386/math98.png)
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A pair of infinitesimals for it
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![infinitesimals := [_xi = x^2*y, _eta = -2*a]](/support/helpjp/helpview.aspx?si=6703/file01386/math107.png)
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The transformation of variables which leaves  invariant
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Note the introduction of _alpha, representing the parameter of the Lie group. Now, to check the invariance of  under this group, you can change variables as follows:
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The change of variables
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![ODE_s := dchange(itr, ODE_y, [r, s(r)])](/support/helpjp/helpview.aspx?si=6703/file01386/math137.png)
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As can be seen above, we arrived at the original  just changing x, y by r, s (this is the meaning of "leaving the ODE invariant").
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