Link to actual problem [6887] \[ \boxed {x {y^{\prime }}^{2}+\left (k -x -y\right ) y^{\prime }+y=0} \]
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 {y}{2}-\frac {k}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {k -x +y}{\sqrt {y}}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {3 x}{2}-\frac {y}{2}+\frac {k}{2}, \underline {\hspace {1.25 ex}}\eta &= -k +x\right ] \\ \left [R &= -\frac {2 k +2 x -y}{9 k^{2}+6 k x -6 k y+x^{2}-2 x y+y^{2}}, S \left (R \right ) &= \ln \left (-k +x \right )-2 \,\operatorname {arctanh}\left (\sqrt {\frac {k^{2}+6 k x -2 k y+9 x^{2}-6 x y+y^{2}}{9 k^{2}+6 k x -6 k y+x^{2}-2 x y+y^{2}}}\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -x^{2}+x y +\frac {3}{2} k x +\frac {1}{2} k y -\frac {1}{2} k^{2}, \underline {\hspace {1.25 ex}}\eta &= -y \left (x -y \right )\right ] \\ \left [R &= \frac {4 y^{2}-8 k y-8 x y+4 k^{2}-8 k x +4 x^{2}}{k^{2}-2 k x -6 k y+x^{2}+6 x y+9 y^{2}}, S \left (R \right ) &= -\frac {4 \left (-\ln \left (-k +4 y\right )+\ln \left (y\right )\right )}{k}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x^{3}-2 x^{2} y +x \,y^{2}-3 x^{2} k +k x y +\frac {5}{2} x \,k^{2}+\frac {1}{2} k^{2} y -\frac {1}{2} k^{3}, \underline {\hspace {1.25 ex}}\eta &= -y \left (2 k x -x^{2}+2 x y -y^{2}\right )\right ] \\ \operatorname {FAIL} \\ \end{align*}