Link to actual problem [9689] \[ \boxed {y^{\prime \prime } x^{2} \left (x^{2}-1\right )-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 x^{2} a +n \left (1+n \right ) \left (x^{2}-1\right )\right ) y=0} \]
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 &= x^{-n} \operatorname {HeunC}\left (0, -n -\frac {1}{2}, -2, -\frac {\left (a +n +1\right ) \left (a -n \right )}{4}, \frac {3}{4}+\frac {\left (a +n \right ) \left (a -1-n \right )}{4}, x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{n} y}{\operatorname {HeunC}\left (0, -n -\frac {1}{2}, -2, -\frac {\left (a +n +1\right ) \left (a -n \right )}{4}, -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{n +1} \operatorname {HeunC}\left (0, n +\frac {1}{2}, -2, -\frac {\left (a +n +1\right ) \left (a -n \right )}{4}, \frac {3}{4}+\frac {\left (a +n \right ) \left (a -1-n \right )}{4}, x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-n} y}{x \operatorname {HeunC}\left (0, n +\frac {1}{2}, -2, -\frac {\left (a +n +1\right ) \left (a -n \right )}{4}, -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right )}\right ] \\ \end{align*}