Link to actual problem [8764] \[ \boxed {a x {y^{\prime }}^{2}-\left (a y+b x -a -b \right ) y^{\prime }+y b=0} \]
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 {1}{2}+\frac {a y +b x -a}{2 b}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {a y-b x +a +b}{\sqrt {y}\, b}, S \left (R \right ) &= \ln \left (y\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= -\frac {a \left (a y -3 b x -a -b \right )}{2 b^{2}}, \underline {\hspace {1.25 ex}}\eta &= x -\frac {a}{b}-1\right ] \\ \left [R &= \frac {a y-2 b x -2 a -2 b}{a b \left (y^{2} a^{2}-2 a b x y+b^{2} x^{2}-6 a^{2} y+6 a b x -6 y a b +6 b^{2} x +9 a^{2}+18 a b +9 b^{2}\right )}, S \left (R \right ) &= \frac {b \left (\ln \left (a y-4 a -4 b \right )-2 \,\operatorname {arctanh}\left (\sqrt {-\frac {y a \left (a y-2 b x -2 a -2 b \right )}{y^{2} a^{2}-2 a b x y+b^{2} x^{2}-6 a^{2} y+6 a b x -6 y a b +6 b^{2} x +9 a^{2}+18 a b +9 b^{2}}+\frac {4 a \left (a y-2 b x -2 a -2 b \right )}{y^{2} a^{2}-2 a b x y+b^{2} x^{2}-6 a^{2} y+6 a b x -6 y a b +6 b^{2} x +9 a^{2}+18 a b +9 b^{2}}+\frac {4 \left (a y-2 b x -2 a -2 b \right ) b}{y^{2} a^{2}-2 a b x y+b^{2} x^{2}-6 a^{2} y+6 a b x -6 y a b +6 b^{2} x +9 a^{2}+18 a b +9 b^{2}}+1}\right )\right )}{a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2}-\frac {2 a b x y -2 b^{2} x^{2}+a^{2} y +3 a b x +a b y +3 b^{2} x -a^{2}-2 a b}{2 b^{2}}, \underline {\hspace {1.25 ex}}\eta &= -\frac {y^{2} a}{b}+x y\right ] \\ \left [R &= \frac {4 y^{2} a^{2}-8 a b x y+4 b^{2} x^{2}-8 a^{2} y-8 y a b -8 a b x -8 b^{2} x +4 a^{2}+8 a b +4 b^{2}}{\left (9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}\right ) a^{2}}, S \left (R \right ) &= \int _{}^{y}-\frac {b}{\textit {\_a} \left (\textit {\_a} a +\frac {\frac {12 \left (y^{2} a^{2}-2 a b x y+b^{2} x^{2}-2 a^{2} y-2 y a b -2 a b x -2 b^{2} x +a^{2}+2 a b +b^{2}\right ) a \textit {\_a}}{9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}}+4 \textit {\_a} a +4 \sqrt {a \textit {\_a} \left (\frac {16 \left (y^{2} a^{2}-2 a b x y+b^{2} x^{2}-2 a^{2} y-2 y a b -2 a b x -2 b^{2} x +a^{2}+2 a b +b^{2}\right ) a \textit {\_a}}{9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}}-\frac {4 \left (y^{2} a^{2}-2 a b x y+b^{2} x^{2}-2 a^{2} y-2 y a b -2 a b x -2 b^{2} x +a^{2}+2 a b +b^{2}\right ) a}{9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}}-\frac {4 \left (y^{2} a^{2}-2 a b x y+b^{2} x^{2}-2 a^{2} y-2 y a b -2 a b x -2 b^{2} x +a^{2}+2 a b +b^{2}\right ) b}{9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}}+4 a +4 b \right )}}{\frac {4 y^{2} a^{2}-8 a b x y+4 b^{2} x^{2}-8 a^{2} y-8 y a b -8 a b x -8 b^{2} x +4 a^{2}+8 a b +4 b^{2}}{9 y^{2} a^{2}+6 a b x y+b^{2} x^{2}-6 a^{2} y-2 a b x -6 y a b -2 b^{2} x +a^{2}+2 a b +b^{2}}-4}-a -b \right )}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2}-\frac {-2 a^{2} b x \,y^{2}+4 a \,b^{2} x^{2} y -2 b^{3} x^{3}+2 a^{2} b x y +2 a \,b^{2} x^{2}+2 a \,b^{2} x y +2 b^{3} x^{2}+a^{3} y +a^{2} b x +2 a^{2} b y +2 a \,b^{2} x +a \,b^{2} y +b^{3} x -a^{3}-3 a^{2} b -3 a \,b^{2}}{2 b^{3}}, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (a^{2} y^{2}-2 a b x y +b^{2} x^{2}-2 a^{2} y -2 a b y \right )}{b^{2}}\right ] \\ \operatorname {FAIL} \\ \end{align*}