Link to actual problem [8067] \[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y=0} \]
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 {KummerU}\left (-2, 1, \frac {x^{2}}{4}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {KummerU}\left (-2, 1, \frac {x^{2}}{4}\right )}\right ] \\ \end{align*}