The infgen command receives a pair of infinitesimals; k, the order of the required prolongation; and the dependent variable, say y(x). It returns the k-extension of the infinitesimal generator (see eta_k and symgen).

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 infgen(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[infgen](..).

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

$X≔\left[-y\,x\right]$

$X≔\left[-y\,x\right]$

$\mathrm{infgen}\left(X\,1\,y\left(x\right)\right)$

$\mathrm{\_F1}\→-y\left(\frac{\partial }{\partial x}\mathrm{\_F1}\right)\+x\left(\frac{\partial }{\partial y}\mathrm{\_F1}\right)\+\left({\mathrm{\_y1}}^2\+1\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\_F1}\right)$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{x+\tan \left(\mathrm{\_F1}\left(x^2+{y\left(x\right)}^2\right)\right)y\left(x\right)}{-y\left(x\right)+x\tan \left(\mathrm{\_F1}\left(x^2+{y\left(x\right)}^2\right)\right)}$

$\mathrm{ODE}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{x+\tan \left(\mathrm{\_F1}\left(x^2+{y\left(x\right)}^2\right)\right)y\left(x\right)}{-y\left(x\right)+x\tan \left(\mathrm{\_F1}\left(x^2+{y\left(x\right)}^2\right)\right)}$

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

$0$

$\mathrm{infgen}\left(X\,1\,\mathrm{ODE}\right)$

$\mathrm{\_F2}\→-y\left(\frac{\partial }{\partial x}\mathrm{\_F2}\right)\+x\left(\frac{\partial }{\partial y}\mathrm{\_F2}\right)\+\frac{\left(\left(1\+{\tan \left(\mathrm{\_F1}\left(x^2\+y^2\right)\right)}^2\right)y^2\+{\tan \left(\mathrm{\_F1}\left(x^2\+y^2\right)\right)}^2{x}^2\+x^2\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\_F2}\right)}{{\left(-y\+x\tan \left(\mathrm{\_F1}\left(x^2\+y^2\right)\right)\right)}^2}$

$X≔\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$

$X≔\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$

$\mathrm{infgen}\left(X\,1\,y\left(x\right)\right)$

$\mathrm{\_F1}\→\mathrm{\ξ}\left(x\,y\right)\left(\frac{\partial }{\partial x}\mathrm{\_F1}\right)\+\mathrm{\η}\left(x\,y\right)\left(\frac{\partial }{\partial y}\mathrm{\_F1}\right)\+\left(\frac{\partial }{\partial x}\mathrm{\η}\left(x\,y\right)\+\left(-\mathrm{\_y1}\left(\frac{\partial }{\partial y}\mathrm{\ξ}\left(x\,y\right)\right)\+\frac{\partial }{\partial y}\mathrm{\η}\left(x\,y\right)-\left(\frac{\partial }{\partial x}\mathrm{\ξ}\left(x\,y\right)\right)\right)\mathrm{\_y1}\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\_F1}\right)$

$X≔\left[\mathrm{\xi }\left(x\,y\,\mathrm{\_y1}\right)\,\mathrm{\eta }\left(x\,y\,\mathrm{\_y1}\right)\right]$

$X≔\left[\mathrm{\xi }\left(x\,y\,\mathrm{\_y1}\right)\,\mathrm{\eta }\left(x\,y\,\mathrm{\_y1}\right)\right]$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=F\left(x\,y\left(x\right)\,\mathrm{diff}\left(y\left(x\right)\,x\right)\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=F\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{infgen}\left(X\,1\,\mathrm{ODE}\right)$

$\mathrm{\_F1}\→\mathrm{\ξ}\left(x\,y\,\mathrm{\_y1}\right)\left(\frac{\partial }{\partial x}\mathrm{\_F1}\right)\+\mathrm{\η}\left(x\,y\,\mathrm{\_y1}\right)\left(\frac{\partial }{\partial y}\mathrm{\_F1}\right)\+\left(-\left(\frac{\partial }{\partial y}\mathrm{\ξ}\left(x\,y\,\mathrm{\_y1}\right)\right){\mathrm{\_y1}}^2\+\left(-F\left(x\,y\,\mathrm{\_y1}\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\ξ}\left(x\,y\,\mathrm{\_y1}\right)\right)\+\frac{\partial }{\partial y}\mathrm{\η}\left(x\,y\,\mathrm{\_y1}\right)-\left(\frac{\partial }{\partial x}\mathrm{\ξ}\left(x\,y\,\mathrm{\_y1}\right)\right)\right)\mathrm{\_y1}\+F\left(x\,y\,\mathrm{\_y1}\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\η}\left(x\,y\,\mathrm{\_y1}\right)\right)\+\frac{\partial }{\partial x}\mathrm{\η}\left(x\,y\,\mathrm{\_y1}\right)\right)\left(\frac{\partial }{\partial \mathrm{\_y1}}\mathrm{\_F1}\right)$