Link to actual problem [9698] \[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
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 &= \operatorname {LegendreP}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreP}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {LegendreQ}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )}\right ] \\ \end{align*}