Link to actual problem [9603] \[ \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y=0} \]
type detected by program
{"second_order_bessel_ode"}
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 {\operatorname {WhittakerM}\left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {1+m}\, \sqrt {m -1}}{2}, x\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerM}\left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {1+m}\, \sqrt {m -1}}{2}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {WhittakerW}\left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {1+m}\, \sqrt {m -1}}{2}, x\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerW}\left (l -\frac {m}{2}+\frac {1}{2}, \frac {\sqrt {1+m}\, \sqrt {m -1}}{2}, x\right )}\right ] \\ \end{align*}