Link to actual problem [9686] \[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \]
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 &= x^{-v} \operatorname {hypergeom}\left (\left [-\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}-v \right ], x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{v} y}{\operatorname {hypergeom}\left (\left [-\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}-v \right ], x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{v +1} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}+v \right ], x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-v} y}{x \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}+v \right ], x^{2}\right )}\right ] \\ \end{align*}