Xchange - Maple Help

DEtools

 Xchange
 change variables in an ODE point symmetry generator

 Calling Sequence Xchange(tr, X, y(x), [t, u(t)])

Parameters

 tr - set of transformation equations of the form $\left\{x=..,y\left(x\right)=..\right\}$ from the old variables on the left hand side to the new variables on the right hand side X - list of coefficients of a point symmetry generator (infinitesimals) as in [xi(x, y), eta(x, y)] y(x) - 'dependent variable' of the problem (it can be any indeterminate function of one variable) [t, u(t)] - (optional) new independent and dependent variables (required when they cannot be inferred from the transformation)

Description

 • 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](..).

Examples

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

Note that the infinitesimals $\left[\mathrm{\xi },\mathrm{\eta }\right]$ can be expressed in terms of $y\left(x\right)$, but also directly in terms of $y$:

 > $X≔\left[-y,x\right]$
 ${X}{≔}\left[{-}{y}{,}{x}\right]$ (1)
 > $\mathrm{tr}≔\left\{x=u\left(t\right)\mathrm{cos}\left(t\right),y\left(x\right)=u\left(t\right)\mathrm{sin}\left(t\right)\right\}$
 ${\mathrm{tr}}{≔}\left\{{x}{=}{u}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){,}{y}{}\left({x}\right){=}{u}{}\left({t}\right){}{\mathrm{sin}}{}\left({t}\right)\right\}$ (2)
 > $\mathrm{Xchange}\left(\mathrm{tr},X,y\left(x\right)\right)$
 $\left[{1}{,}{0}\right]$ (3)
 > $X≔\left[\frac{1}{x+y},1\right]$
 ${X}{≔}\left[\frac{{1}}{{x}{+}{y}}{,}{1}\right]$ (4)
 > $\mathrm{tr}≔\left\{x=-\mathrm{LambertW}\left(-r\mathrm{exp}\left(-s\left(r\right)-1\right)\right)-s\left(r\right)-1,y\left(x\right)=s\left(r\right)\right\}$
 ${\mathrm{tr}}{≔}\left\{{x}{=}{-}{\mathrm{LambertW}}{}\left({-}{r}{}{{ⅇ}}^{{-}{s}{}\left({r}\right){-}{1}}\right){-}{s}{}\left({r}\right){-}{1}{,}{y}{}\left({x}\right){=}{s}{}\left({r}\right)\right\}$ (5)
 > $\mathrm{Xchange}\left(\mathrm{tr},X,y\left(x\right)\right)$
 $\left[{0}{,}{1}\right]$ (6)

This example illustrates a more general change of variables where the new variables are $\left[v,u\left(v\right)\right]$:

 > $X≔\left[x,y\right]$
 ${X}{≔}\left[{x}{,}{y}\right]$ (7)
 > $\mathrm{tr}≔\left\{x=\mathrm{\phi }\left(u,v\left(u\right)\right),y\left(x\right)=\mathrm{\psi }\left(u,v\left(u\right)\right)\right\}$
 ${\mathrm{tr}}{≔}\left\{{x}{=}{\mathrm{\phi }}{}\left({u}{,}{v}{}\left({u}\right)\right){,}{y}{}\left({x}\right){=}{\mathrm{\psi }}{}\left({u}{,}{v}{}\left({u}\right)\right)\right\}$ (8)
 > $\mathrm{Xchange}\left(\mathrm{tr},X,y\left(x\right),\left[u,v\left(u\right)\right]\right)$
 $\left[\frac{{\mathrm{\phi }}{}\left({u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right){-}{\mathrm{\psi }}{}\left({u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right)}{\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right){-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right)}{,}{-}\frac{{\mathrm{\phi }}{}\left({u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right){-}{\mathrm{\psi }}{}\left({u}{,}{v}\right){}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right)}{\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right){-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\phi }}{}\left({u}{,}{v}\right)\right){}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\psi }}{}\left({u}{,}{v}\right)\right)}\right]$ (9)