Link to actual problem [8763] \[ \boxed {a x {y^{\prime }}^{2}+\left (b x -a y+c \right ) y^{\prime }-y b=0} \]
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 {x \left (a^{2} y^{2}+2 a b x y +b^{2} x^{2}+2 c b x +c^{2}\right )}{c^{2}}, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (a^{2} y^{2}+2 a b x y +b^{2} x^{2}-a c y +c b x \right )}{c^{2}}\right ] \\ \operatorname {FAIL} \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x \left (a^{2} y^{2}+2 a b x y +b^{2} x^{2}-a c y +c b x \right )}{c^{2}}, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (a^{2} y^{2}+2 a b x y +b^{2} x^{2}-2 a c y +c^{2}\right )}{c^{2}}\right ] \\ \operatorname {FAIL} \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {a x \left (3 a^{2} y^{2}+6 a b x y +3 b^{2} x^{2}-a c y +5 c b x \right )}{c^{3}}, \underline {\hspace {1.25 ex}}\eta &= 1+\frac {3 a^{3} y^{3}+6 a^{2} b x \,y^{2}+3 a \,b^{2} x^{2} y -4 a^{2} c \,y^{2}+2 a b c x y +b \,c^{2} x}{c^{3}}\right ] \\ \operatorname {FAIL} \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {3 a^{2} b x \,y^{2}+6 a \,b^{2} x^{2} y +3 b^{3} x^{3}-2 a b c x y +4 b^{2} c \,x^{2}+a \,c^{2} y -c^{3}}{c^{2} a}, \underline {\hspace {1.25 ex}}\eta &= \frac {b y \left (3 a^{2} y^{2}+6 a b x y +3 b^{2} x^{2}-5 a c y +c b x \right )}{c^{2} a}\right ] \\ \operatorname {FAIL} \\ \end{align*}