Link to actual problem [4323] \[ \boxed {a \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y=0} \]
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 &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {a^{2}-y^{2}}{x^{2}}, S \left (R \right ) &= \frac {\ln \left (y-a \right )-\ln \left (y+a \right )}{2 a}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {-a^{2}+x^{2}}}, S \left (R \right ) &= \frac {\ln \left (x -a \right )-\ln \left (x +a \right )}{2 a}\right ] \\ \end{align*}