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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

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $\mathrm{ODE_y}≔\mathrm{diff}\left(y\left(x\right),x\right)=\frac{2a}{-{x}^{2}y\left(x\right)+{x}^{2}\cdot 2F\left(\frac{x{y\left(x\right)}^{2}-4a}{x}\right)a}$
 ${\mathrm{ODE_y}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{2}{}{a}}{{-}{{x}}^{{2}}{}{y}{}\left({x}\right){+}{2}{}{{x}}^{{2}}{}{F}{}\left(\frac{{x}{}{{y}{}\left({x}\right)}^{{2}}{-}{4}{}{a}}{{x}}\right){}{a}}$ (1)
 > $\mathrm{odeadvisor}\left(\mathrm{ODE_y}\right)$
 $\left[\left[{\mathrm{_1st_order}}{,}{\mathrm{_with_symmetry_\left[F\left(x\right),G\left(y\right)\right]}}\right]\right]$ (2)

A pair of infinitesimals for it

 > $\mathrm{infinitesimals}≔\mathrm{symgen}\left(\mathrm{ODE_y}\right)$
 ${\mathrm{infinitesimals}}{≔}\left[{\mathrm{_ξ}}{=}{{x}}^{{2}}{}{y}{,}{\mathrm{_η}}{=}{-}{2}{}{a}\right]$ (3)

The transformation of variables which leaves $\mathrm{ODE_y}$ invariant

 > $\mathrm{tr}≔\mathrm{transinv}\left(\mathrm{infinitesimals},y\left(x\right),s\left(r\right)\right)$
 ${\mathrm{tr}}{≔}\left\{{r}{=}\frac{{x}}{{{\mathrm{_α}}}^{{2}}{}{a}{}{x}{-}{\mathrm{_α}}{}{x}{}{y}{}\left({x}\right){+}{1}}{,}{s}{}\left({r}\right){=}{-}{2}{}{\mathrm{_α}}{}{a}{+}{y}{}\left({x}\right)\right\}$ (4)

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

 > $\mathrm{itr}≔\mathrm{solve}\left(\mathrm{tr},\left\{x,y\left(x\right)\right\}\right)$
 ${\mathrm{itr}}{≔}\left\{{x}{=}\frac{{r}}{{1}{+}{{\mathrm{_α}}}^{{2}}{}{a}{}{r}{+}{s}{}\left({r}\right){}{\mathrm{_α}}{}{r}}{,}{y}{}\left({x}\right){=}{s}{}\left({r}\right){+}{2}{}{\mathrm{_α}}{}{a}\right\}$ (5)

The change of variables

 > $\mathrm{ODE_s}≔\mathrm{dchange}\left(\mathrm{itr},\mathrm{ODE_y},\left[r,s\left(r\right)\right]\right):$
 > $\mathrm{diff}\left(s\left(r\right),r\right)=\mathrm{solve}\left(\mathrm{ODE_s},\mathrm{diff}\left(s\left(r\right),r\right)\right)$
 $\frac{{ⅆ}}{{ⅆ}{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{s}{}\left({r}\right){=}\frac{{2}{}{a}}{{{r}}^{{2}}{}\left({2}{}{F}{}\left(\frac{{{s}{}\left({r}\right)}^{{2}}{}{r}{-}{4}{}{a}}{{r}}\right){}{a}{-}{s}{}\left({r}\right)\right)}$ (6)

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