Link to actual problem [4114] \[ \boxed {\left (x +1\right ) {y^{\prime }}^{2}-\left (y+x \right ) y^{\prime }+y=0} \]
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 {y}{2}+\frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {x -y}{\sqrt {y}}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 2+\frac {3 x}{2}-\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= x\right ] \\ \left [R &= \frac {4+2 x -y}{x^{2}-2 x y+y^{2}+8 x -8 y+16}, S \left (R \right ) &= \ln \left (x \right )-2 \,\operatorname {arctanh}\left (\sqrt {\frac {4 x \left (4+2 x -y\right )}{x^{2}-2 x y+y^{2}+8 x -8 y+16}+1}\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}-2 x y-4 y+x^{2}}{y^{2}}, S \left (R \right ) &= \frac {\frac {y^{2}-2 x y-4 y+x^{2}}{y}+4}{2 \sqrt {x^{2}-2 x y+y^{2}}}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \end{align*}