Link to actual problem [4423] \[ \boxed {y^{\prime } y-y^{2}+y^{2} {y^{\prime }}^{2}=x} \]
type detected by program
{"unknown"}
type detected by Maple
[_rational, [_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 {1}{2 y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x +\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= y +\frac {x}{2 y}\right ] \\ \left [R &= \frac {1+4 y^{2}+4 x}{16 x^{2}+16 x +4}, S \left (R \right ) &= \ln \left (2 x +1\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2} x^{2}+\frac {1}{2} y^{2}+\frac {1}{8}+\frac {3}{4} x, \underline {\hspace {1.25 ex}}\eta &= \frac {x \left (8 y^{2}+6 x +1\right )}{8 y}\right ] \\ \operatorname {FAIL} \\ \end{align*}