Link to actual problem [7583] \[ \boxed {x^{2} \left (4+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y=0} \]
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}}}{\left (x +4\right )^{\frac {9}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +4\right )^{\frac {9}{4}} y}{x^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [1, 3\right ], \left [\frac {7}{4}\right ], -\frac {x}{4}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [1, 3\right ], \left [\frac {7}{4}\right ], -\frac {x}{4}\right ) x}\right ] \\ \end{align*}