Link to actual problem [8317] \[ \boxed {y^{\prime \prime }-\left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (x -1\right )^{2}}+\frac {3}{16 x \left (x -1\right )}\right ) y=0} \]
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 &= \sqrt {-1+x}\, x^{\frac {1}{4}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {-1+x}\, x^{\frac {1}{4}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {-1+x}\, x^{\frac {1}{4}} \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {-1+x}\, x^{\frac {1}{4}} \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {x}\right )}\right ] \\ \end{align*}