Link to actual problem [8793] \[ \boxed {{\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x} \sqrt {-{\mathrm e}^{-2 x -2 y}+{\mathrm e}^{-2 x}+{\mathrm e}^{-2 y}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} {\mathrm e}^{-y} \sqrt {{\mathrm e}^{-2 x} \left ({\mathrm e}^{2 x}+{\mathrm e}^{2 y}-1\right )}\, \ln \left (\left (\sqrt {{\mathrm e}^{-2 x} \left ({\mathrm e}^{2 x}+{\mathrm e}^{2 y}-1\right )}\, {\mathrm e}^{2 x} \sqrt {{\mathrm e}^{-2 x}}+{\mathrm e}^{y}\right ) \sqrt {{\mathrm e}^{-2 x}}\right )}{\sqrt {\left ({\mathrm e}^{2 x}+{\mathrm e}^{2 y}-1\right ) {\mathrm e}^{-2 x} {\mathrm e}^{-2 y}}\, \sqrt {{\mathrm e}^{-2 x}}}\right ] \\ \end{align*}