The Xchange command receives a change of variables transformation set and a pair of infinitesimals $\left[\mathrm{\xi }\,\mathrm{\eta }\right]$, representing the coefficients of a point symmetry generator of an ODE, and the dependent variable y(x), and returns the symmetry in the new variables. The change of variables is performed by making calls to dchange, and hence the same extra arguments accepted by dchange are accepted by Xchange as well. This change of variables takes into account that $\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$ are the coefficients of a differential operator $\mathrm{expr}$. Thus, if the new variables are $\left\{t\,u\left(t\right)\right\}$, then the returned list is constituted by the coefficients of the differential operator $\mathrm{expr}$, where $\mathrm{\ξ2}\left(t\,u\left(t\right)\right)$ and $\mathrm{\η2}\left(t\,u\left(t\right)\right)$ also incorporate any factor that appears when changing variables in $\frac{d}{\mathrm{dx}}$ and $\frac{d}{\mathrm{dy}}$.

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

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$X≔\left[-y\,x\right]$

$X≔\left[-y\,x\right]$

$\mathrm{tr}≔\left\{x=u\left(t\right)\cos \left(t\right)\,y\left(x\right)=u\left(t\right)\sin \left(t\right)\right\}$

$\mathrm{tr}≔\left\{x=u\left(t\right)\cos \left(t\right)\,y\left(x\right)=u\left(t\right)\sin \left(t\right)\right\}$

$\mathrm{Xchange}\left(\mathrm{tr}\,X\,y\left(x\right)\right)$

$\left[1\,0\right]$

$X≔\left[\frac{1}{x+y}\,1\right]$

$X≔\left[\frac{1}{x+y}\,1\right]$

$\mathrm{tr}≔\left\{x=-\mathrm{LambertW}\left(-r\exp \left(-s\left(r\right)-1\right)\right)-s\left(r\right)-1\,y\left(x\right)=s\left(r\right)\right\}$

$\mathrm{tr}≔\left\{x=-\mathrm{LambertW}\left(-r{\ⅇ}^{-s\left(r\right)-1}\right)-s\left(r\right)-1\,y\left(x\right)=s\left(r\right)\right\}$

$\mathrm{Xchange}\left(\mathrm{tr}\,X\,y\left(x\right)\right)$

$\left[0\,1\right]$

$X≔\left[x\,y\right]$

$X≔\left[x\,y\right]$

$\mathrm{tr}≔\left\{x=\mathrm{\phi }\left(u\,v\left(u\right)\right)\,y\left(x\right)=\mathrm{\psi }\left(u\,v\left(u\right)\right)\right\}$

$\mathrm{tr}≔\left\{x=\mathrm{\phi }\left(u\,v\left(u\right)\right)\,y\left(x\right)=\mathrm{\psi }\left(u\,v\left(u\right)\right)\right\}$

$\mathrm{Xchange}\left(\mathrm{tr}\,X\,y\left(x\right)\,\left[u\,v\left(u\right)\right]\right)$

$\left[\frac{\mathrm{\phi }\left(u\,v\right)\left(\frac{\partial }{\partial v}\mathrm{\psi }\left(u\,v\right)\right)-\mathrm{\psi }\left(u\,v\right)\left(\frac{\partial }{\partial v}\mathrm{\phi }\left(u\,v\right)\right)}{\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(u\,v\right)\right)\left(\frac{\partial }{\partial v}\mathrm{\psi }\left(u\,v\right)\right)-\left(\frac{\partial }{\partial v}\mathrm{\phi }\left(u\,v\right)\right)\left(\frac{\partial }{\partial u}\mathrm{\psi }\left(u\,v\right)\right)}\,-\frac{\mathrm{\phi }\left(u\,v\right)\left(\frac{\partial }{\partial u}\mathrm{\psi }\left(u\,v\right)\right)-\mathrm{\psi }\left(u\,v\right)\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(u\,v\right)\right)}{\left(\frac{\partial }{\partial u}\mathrm{\phi }\left(u\,v\right)\right)\left(\frac{\partial }{\partial v}\mathrm{\psi }\left(u\,v\right)\right)-\left(\frac{\partial }{\partial v}\mathrm{\phi }\left(u\,v\right)\right)\left(\frac{\partial }{\partial u}\mathrm{\psi }\left(u\,v\right)\right)}\right]$