Link to actual problem [4322] \[ \boxed {\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+x y^{\prime }-y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= -\frac {b^{2} y}{a^{2}}, \underline {\hspace {1.25 ex}}\eta &= x\right ] \\ \left [R &= \frac {y^{2} b^{2}+a^{2} x^{2}}{b^{2}}, S \left (R \right ) &= -\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {y^{2} b^{2}}}\right )}{b \sqrt {a^{2}}}\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 {-b^{2}+x^{2}}}, S \left (R \right ) &= -\frac {\ln \left (x +b \right )-\ln \left (-b +x \right )}{2 b}\right ] \\ \end{align*}