Link to actual problem [7592] \[ \boxed {4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y=0} \]
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 {\operatorname {LegendreP}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}{x^{\frac {3}{8}} \sqrt {x^{2}-1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{8}} \sqrt {x^{2}-1}\, y}{\operatorname {LegendreP}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {LegendreQ}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}{x^{\frac {3}{8}} \sqrt {x^{2}-1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{8}} \sqrt {x^{2}-1}\, y}{\operatorname {LegendreQ}\left (-\frac {3}{8}, \frac {5}{8}, \sqrt {-x^{2}+1}\right )}\right ] \\ \end{align*}