Link to actual problem [11216] \[ \boxed {4 \,{\mathrm e}^{2 y} {y^{\prime }}^{2}+2 y^{\prime } x=1} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {x \,{\mathrm e}^{-2 y}}{4}\right ] \\ \left [R &= -\frac {x^{2}}{2}-2 \,{\mathrm e}^{2 y}, S \left (R \right ) &= x\right ] \\ \end{align*}