Link to actual problem [9546] \[ \boxed {x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y=0} \]
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 &= \sec \left (x \right ) \operatorname {BesselJ}\left (\sqrt {a}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sec \left (x \right ) \operatorname {BesselJ}\left (\sqrt {a}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sec \left (x \right ) \operatorname {BesselY}\left (\sqrt {a}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sec \left (x \right ) \operatorname {BesselY}\left (\sqrt {a}, x\right )}\right ] \\ \end{align*}