Link to actual problem [9688] \[ \boxed {y^{\prime \prime }-\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (a \left (1+a \right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode", "second_order_change_of_variable_on_y_method_2"}
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*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-a} y}{x \left (2 x^{2} a +x^{2}-2 a -3\right )}\right ] \\ \end{align*}