Link to actual problem [8331] \[ \boxed {y^{\prime \prime }+\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}=0} \]
type detected by program
{"kovacic"}
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 &= \sqrt {x^{2}-1}\, \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x^{2}-1}\, \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x^{2}-1}\, \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x^{2}-1}\, \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, x\right )}\right ] \\ \end{align*}