The Xgauge command receives a list of a pair of infinitesimals of a one-parameter Lie group, the dependent variable y(x), and optionally the "gauge" in the form of $\mathrm{\xi }=\mathrm{expr}$ or $\mathrm{\eta }=\mathrm{expr}$. It returns the given infinitesimals "gauged" to satisfy the indication received. If no gauge is given, the command returns the given generator rewritten in the most general form by introducing an arbitrary function.

This command also works with dynamical symmetries, in which case the ODE assumed to be invariant under the given infinitesimals is also required as an argument. The right hand side of the given nth order ODE is then used to replace the nth order derivatives of the dependent variable appearing in the infinitesimal generator.

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

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

$X≔\left[x\,x+y\right]$

$X≔\left[x\,x+y\right]$

$\mathrm{ODE}≔\mathrm{equinv}\left(X\,y\left(x\right)\right)$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\ln \left(x\right)+\mathrm{f\_\_1}\left(-\frac{x\ln \left(x\right)-y\left(x\right)}{x}\right)$

$\mathrm{symtest}\left(X\,\mathrm{ODE}\right)$

$0$

$\mathrm{Xgauge}\left(X\,y\left(x\right)\,\mathrm{\xi }=0\right)$

$\left[0\,-\mathrm{\_y1}x+x+y\right]$

$\mathrm{`X\_xi=0`}≔\mathrm{Xgauge}\left(X\,\mathrm{ODE}\,\mathrm{\xi }=0\right)$

$\mathrm{X\_xi=0}≔\left[0\,y+\left(-\mathrm{f\_\_1}\left(-\frac{x\ln \left(x\right)-y}{x}\right)-\ln \left(x\right)+1\right)x\right]$

$\mathrm{symtest}\left(\mathrm{`X\_xi=0`}\,\mathrm{ODE}\right)$

$0$

$X$

$\left[x\,x+y\right]$

$\mathrm{X\_general}≔\mathrm{Xgauge}\left(X\,\mathrm{ODE}\right)$

$\mathrm{X\_general}≔\left[\mathrm{\_F2}\left(x\,y\right)\,y+\left(-\mathrm{f\_\_1}\left(-\frac{x\ln \left(x\right)-y}{x}\right)-\ln \left(x\right)+1\right)x+\mathrm{\_F2}\left(x\,y\right)\mathrm{f\_\_1}\left(-\frac{x\ln \left(x\right)-y}{x}\right)+\mathrm{\_F2}\left(x\,y\right)\ln \left(x\right)\right]$

$\mathrm{symtest}\left(\mathrm{X\_general}\,\mathrm{ODE}\right)$

$0$