Link to actual problem [9699] \[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (\left (x^{2}-1\right ) \left (x^{2} a +b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
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 &= {\mathrm e}^{x \sqrt {-a}} \operatorname {HeunC}\left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) \left (x^{2}-1\right )^{\frac {k}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x \sqrt {-a}} \left (x^{2}-1\right )^{-\frac {k}{2}} y}{\operatorname {HeunC}\left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x \sqrt {-a}} \operatorname {HeunC}\left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) \left (1+x \right )^{-\frac {k}{2}} \left (-1+x \right )^{\frac {k}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x \sqrt {-a}} \left (1+x \right )^{\frac {k}{2}} \left (-1+x \right )^{-\frac {k}{2}} y}{\operatorname {HeunC}\left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}