Link to actual problem [10089] \[ \boxed {x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y^{\prime } y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {b^{2}}{x}-x, \underline {\hspace {1.25 ex}}\eta &= -\ln \left (y \right ) y\right ] \\ \left [R &= \frac {\sqrt {-b^{2}+x^{2}}}{\ln \left (y\right )}, S \left (R \right ) &= -\frac {\ln \left (-b^{2}+x^{2}\right )}{2}\right ] \\ \end{align*}