Link to actual problem [4115] \[ \boxed {\left (a -x \right ) {y^{\prime }}^{2}+y^{\prime } y=b} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Clairaut]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y^{2}-4 b x, S \left (R \right ) &= \frac {y}{2 b}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {a -x}}, S \left (R \right ) &= \frac {\ln \left (x -a \right )}{2}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}+4 a b -4 b x}{x^{2}}, S \left (R \right ) &= -\frac {\ln \left (\frac {-8 a b +4 b x +4 \sqrt {-a b}\, y}{x}\right )}{2 \sqrt {-a b}}\right ] \\ \end{align*}
\begin{align*} \\ \text {Expression too large to display} \\ \end{align*}