Link to actual problem [15135] \[ \boxed {y-x y^{\prime }-\sqrt {a^{2} {y^{\prime }}^{2}+b^{2}}=0} \]
type detected by program
{"clairaut"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
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 {a^{2} y}{b^{2}}, \underline {\hspace {1.25 ex}}\eta &= x\right ] \\ \left [R &= \frac {y^{2} a^{2}+b^{2} x^{2}}{a^{2}}, S \left (R \right ) &= -\frac {b^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {y^{2} a^{2}}}\right )}{a \sqrt {b^{2}}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {b^{2}-y^{2}}{x^{2}}, S \left (R \right ) &= -\frac {\ln \left (b +y\right )-\ln \left (y-b \right )}{2 b}\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*}