Link to actual problem [11225] \[ \boxed {y y^{\prime }-\left (x -b \right ) {y^{\prime }}^{2}=a} \]
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 &= -4 x a +y^{2}, S \left (R \right ) &= \frac {y}{2 a}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {b -x}}, S \left (R \right ) &= \frac {\ln \left (x -b \right )}{2}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y^{2}+4 a b -4 x a}{x^{2}}, S \left (R \right ) &= -\frac {\ln \left (\frac {-8 a b +4 x a +4 \sqrt {-a b}\, y}{x}\right )}{2 \sqrt {-a b}}\right ] \\ \end{align*}
\begin{align*} \\ \text {Expression too large to display} \\ \end{align*}