Link to actual problem [4100] \[ \boxed {x {y^{\prime }}^{2}+\left (a +x -y\right ) y^{\prime }-y=0} \]
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 {a}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {a +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 {a}{2}-\frac {3 x}{2}-\frac {y}{2}, \underline {\hspace {1.25 ex}}\eta &= x +a\right ] \\ \left [R &= \frac {2 a -2 x -y}{9 a^{2}-6 x a -6 a y+x^{2}+2 x y+y^{2}}, S \left (R \right ) &= -\ln \left (x +a \right )-2 \,\operatorname {arctanh}\left (\sqrt {\frac {a^{2}-6 x a -2 a y+9 x^{2}+6 x y+y^{2}}{9 a^{2}-6 x a -6 a 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} x a -\frac {1}{2} a y +\frac {1}{2} a^{2}, \underline {\hspace {1.25 ex}}\eta &= y \left (x +y \right )\right ] \\ \left [R &= \frac {4 y^{2}-8 a y+8 x y+4 a^{2}+8 x a +4 x^{2}}{a^{2}+2 x a -6 a y+x^{2}-6 x y+9 y^{2}}, S \left (R \right ) &= -\frac {4 \left (-\ln \left (4 y-a \right )+\ln \left (y\right )\right )}{a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x^{3}+2 x^{2} y +x \,y^{2}+3 x^{2} a +a x y +\frac {5}{2} x \,a^{2}-\frac {1}{2} a^{2} y +\frac {1}{2} a^{3}, \underline {\hspace {1.25 ex}}\eta &= y \left (2 x a +x^{2}+2 x y +y^{2}\right )\right ] \\ \operatorname {FAIL} \\ \end{align*}