Link to actual problem [15114] \[ \boxed {x {y^{\prime }}^{2}-y y^{\prime }-y^{\prime }=-1} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
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 &= \frac {y}{2}+\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= 1\right ] \\ \left [R &= \frac {y^{2}}{4}-x +\frac {y}{2}, S \left (R \right ) &= y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {1}{2}+2 x -\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {-2 y+4 x -1}{4 y^{2}}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2}+x y -3 x +\frac {1}{2} y, \underline {\hspace {1.25 ex}}\eta &= y^{2}-2 x\right ] \\ \left [R &= -\frac {-y^{2}+4 x -2 y-1}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}, S \left (R \right ) &= \int _{}^{y}\frac {1}{\textit {\_a}^{2}+\frac {\left (-\frac {2 \left (-y^{2}+4 x -2 y-1\right ) \textit {\_a}}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}+\sqrt {-\frac {\left (-y^{2}+4 x -2 y-1\right ) \textit {\_a}^{2}}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}+\frac {2 \left (-y^{2}+4 x -2 y-1\right ) \textit {\_a}}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}+\frac {-y^{2}+4 x -2 y-1}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}+1}-\frac {-y^{2}+4 x -2 y-1}{4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1}-1\right ) \left (4 x^{2}-8 x y+4 y^{2}-4 x +4 y+1\right )}{-y^{2}+4 x -2 y-1}}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x \,y^{2}-2 x^{2}-\frac {1}{2}-x y +4 x -\frac {1}{2} y, \underline {\hspace {1.25 ex}}\eta &= y^{3}-3 x y +3 x\right ] \\ \operatorname {FAIL} \\ \end{align*}