Link to actual problem [9417] \[ \boxed {4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y=0} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{\sqrt {\cos \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \sqrt {\cos \left (x \right )}\, y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {-2 \ln \left (i \cos \left (x \right )+\sin \left (x \right )\right )+i \sin \left (2 x \right )}{2 \sqrt {\cos \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \sqrt {\cos \left (x \right )}\, y}{-2 \ln \left (i \cos \left (x \right )+\sin \left (x \right )\right )+i \sin \left (2 x \right )}\right ] \\ \end{align*}