Link to actual problem [8011] \[ \boxed {8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \]
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 &= \frac {x^{\frac {1}{4}}}{\sqrt {x^{2}-1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x^{2}-1}\, y}{x^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {3}{8}} \operatorname {LegendreQ}\left (-\frac {1}{8}, \frac {1}{8}, \sqrt {-x^{2}+1}\right )}{\sqrt {x^{2}-1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x^{2}-1}\, y}{x^{\frac {3}{8}} \operatorname {LegendreQ}\left (-\frac {1}{8}, \frac {1}{8}, \sqrt {-x^{2}+1}\right )}\right ] \\ \end{align*}