Link to actual problem [8762] \[ \boxed {\left (3 x +5\right ) {y^{\prime }}^{2}-\left (3 y+x \right ) y^{\prime }+y=0} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
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 {x}{2}+\frac {3 y}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {-3 y+x}{\sqrt {y}}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 10+\frac {9 x}{2}-\frac {9 y}{2}, \underline {\hspace {1.25 ex}}\eta &= x\right ] \\ \left [R &= \frac {20+6 x -9 y}{27 x^{2}-162 x y+243 y^{2}+360 x -1080 y+1200}, S \left (R \right ) &= \frac {\ln \left (x \right )}{3}-\frac {2 \,\operatorname {arctanh}\left (\sqrt {\frac {12 x \left (20+6 x -9 y\right )}{9 x^{2}-54 x y+81 y^{2}+120 x -360 y+400}+1}\right )}{3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {1}{3} x^{2}+x y +\frac {10}{3} y, \underline {\hspace {1.25 ex}}\eta &= -\frac {y \left (-3 y +x \right )}{3}\right ] \\ \left [R &= \frac {9 y^{2}-6 x y+x^{2}-20 y}{y^{2}}, S \left (R \right ) &= \frac {\frac {3 \left (9 y^{2}-6 x y+x^{2}-20 y\right )}{10 y}+6}{\sqrt {x^{2}-6 x y+9 y^{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{9} x^{3}-\frac {2}{3} x^{2} y +x \,y^{2}-\frac {10}{27} x^{2}-\frac {5}{9} x y +\frac {5}{3} y^{2}+\frac {100}{27} y, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (3 x^{2}-18 x y +27 y^{2}-10 x \right )}{27}\right ] \\ \operatorname {FAIL} \\ \end{align*}