Link to actual problem [4418] \[ \boxed {y-x y^{\prime }-\sqrt {b^{2}-{y^{\prime }}^{2} a^{2}}=0} \]
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 &= y, \underline {\hspace {1.25 ex}}\eta &= \frac {b^{2} x}{a^{2}}\right ] \\ \left [R &= \frac {y^{2} a^{2}-b^{2} x^{2}}{a^{2}}, S \left (R \right ) &= \frac {a \ln \left (\frac {b^{2} x}{\sqrt {b^{2}}}+\sqrt {y^{2} a^{2}}\right )}{\sqrt {b^{2}}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {b^{2}-y^{2}}{x^{2}}, S \left (R \right ) &= \frac {\ln \left (y-b \right )-\ln \left (b +y\right )}{2 b}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {a^{2}+x^{2}}}, S \left (R \right ) &= \frac {\arctan \left (\frac {x}{a}\right )}{a}\right ] \\ \end{align*}