Link to actual problem [7593] \[ \boxed {3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y=0} \]
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 {\sqrt {x}}{\left (-2 x^{2}+4\right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-2 x^{2}+4\right )^{\frac {3}{4}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1\right ], \left [\frac {11}{12}\right ], \frac {x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1\right ], \left [\frac {11}{12}\right ], \frac {x^{2}}{2}\right )}\right ] \\ \end{align*}