Link to actual problem [11309] \[ \boxed {x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y=0} \]
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 &= \operatorname {WhittakerM}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) x^{n} {\mathrm e}^{n x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-n} {\mathrm e}^{-n x} y}{\operatorname {WhittakerM}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {WhittakerW}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right ) x^{n} {\mathrm e}^{n x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-n} {\mathrm e}^{-n x} y}{\operatorname {WhittakerW}\left (\frac {i n^{2}}{\sqrt {a -n}\, \sqrt {a +n}}, \frac {1}{2}, 2 i \sqrt {a -n}\, \sqrt {a +n}\, x \right )}\right ] \\ \end{align*}