Link to actual problem [9357] \[ \boxed {y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{\sec \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sec \left (x \right ) \left (2 \ln \left (\cos \left (x \right )+i \sin \left (x \right )\right )+i \sin \left (2 x \right )\right )}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y}{\sec \left (x \right ) \left (2 \ln \left (\cos \left (x \right )+i \sin \left (x \right )\right )+i \sin \left (2 x \right )\right )}\right ] \\ \end{align*}