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](..).

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

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

$\mathrm{ODE\_y}≔\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{2a}{-x^{2}y\left(x\right)+x^2\cdot 2F\left(\frac{x{y\left(x\right)}^2-4a}{x}\right)a}$

$\mathrm{ODE\_y}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{2a}{-x^{2}y\left(x\right)+2x^{2}F\left(\frac{x{y\left(x\right)}^2-4a}{x}\right)a}$

$\mathrm{odeadvisor}\left(\mathrm{ODE\_y}\right)$

$\left[\left[\mathrm{\_1st\_order}\,\mathrm{\_with\_symmetry\_[F(x),G(y)]}\right]\right]$

$\mathrm{infinitesimals}≔\mathrm{symgen}\left(\mathrm{ODE\_y}\right)$

$\mathrm{infinitesimals}≔\left[\mathrm{\_\ξ}=x^{2}y\,\mathrm{\_\η}=-2a\right]$

$\mathrm{tr}≔\mathrm{transinv}\left(\mathrm{infinitesimals}\,y\left(x\right)\,s\left(r\right)\right)$

$\mathrm{tr}≔\left\{r=\frac{x}{{\mathrm{\_\α}}^{2}ax-\mathrm{\_\α}xy\left(x\right)+1}\,s\left(r\right)=-2\mathrm{\_\α}a+y\left(x\right)\right\}$

$\mathrm{itr}≔\mathrm{solve}\left(\mathrm{tr}\,\left\{x\,y\left(x\right)\right\}\right)$

$\mathrm{itr}≔\left\{x=\frac{r}{1+{\mathrm{\_\α}}^{2}ar+s\left(r\right)\mathrm{\_\α}r}\,y\left(x\right)=s\left(r\right)+2\mathrm{\_\α}a\right\}$

$\mathrm{ODE\_s}≔\mathrm{dchange}\left(\mathrm{itr}\,\mathrm{ODE\_y}\,\left[r\,s\left(r\right)\right]\right)\:$

$\mathrm{diff}\left(s\left(r\right)\,r\right)=\mathrm{solve}\left(\mathrm{ODE\_s}\,\mathrm{diff}\left(s\left(r\right)\,r\right)\right)$

$\frac{\ⅆ}{\ⅆr}s\left(r\right)=\frac{2a}{r^2\left(2F\left(\frac{{s\left(r\right)}^{2}r-4a}{r}\right)a-s\left(r\right)\right)}$