The general form of the Liouville ODE is given by the following:

Liouville_ode := diff(y(x),x,x)+g(y(x))*diff(y(x),x)^2+f(x)*diff(y(x),x) = 0;

$\mathrm{Liouville\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+g\left(y\left(x\right)\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2+f\left(x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0$

where g and f are arbitrary functions. See Goldstein and Braun, "Advanced Methods for the Solution of Differential Equations".

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\,\mathrm{symgen}\,\mathrm{symtest}\right)\:$

$\mathrm{odeadvisor}\left(\mathrm{Liouville\_ode}\right)$

$\left[\mathrm{\_Liouville}\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_x\_y1}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_xy}\right]\right]$

$\mathrm{symmetries}≔\mathrm{symgen}\left(\mathrm{Liouville\_ode}\right)$

$\mathrm{symmetries}≔\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}={\ⅇ}^{-\int g\left(y\right)\ⅆy}\right],\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=\int {\ⅇ}^{\int g\left(y\right)\ⅆy}\ⅆy{\ⅇ}^{-\int g\left(y\right)\ⅆy}\right],\left[\mathrm{\_\ξ}={\ⅇ}^{-\int -f\left(x\right)\ⅆx}\,\mathrm{\_\η}=0\right],\left[\mathrm{\_\ξ}=\int {\ⅇ}^{\int -f\left(x\right)\ⅆx}\ⅆx{\ⅇ}^{-\int -f\left(x\right)\ⅆx}\,\mathrm{\_\η}=0\right]$

$\mathrm{map}\left(\mathrm{symtest}\,\left[\mathrm{symmetries}\right]\,\mathrm{Liouville\_ode}\right)$

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

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Liouville\_ode}\right)$

$\mathrm{ans}≔{\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}{\ⅇ}^{\int g\left(\mathrm{\_b}\right)\ⅆ\mathrm{\_b}}\ⅆ\mathrm{\_b}-\mathrm{c\_\_1}\int {\ⅇ}^{-\int f\left(x\right)\ⅆx}\ⅆx-\mathrm{c\_\_2}=0$

$\mathrm{odetest}\left(\mathrm{ans}\,\mathrm{Liouville\_ode}\right)$

$0$