Link to actual problem [7564] \[ \boxed {\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y=0} \]
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 &= \left (i \sqrt {7}+4 x +1\right )^{-\frac {3}{4}+\frac {3 i \sqrt {7}}{28}} \left (i \sqrt {7}-4 x -1\right )^{-\frac {3}{4}-\frac {3 i \sqrt {7}}{28}} \left (1+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (i \sqrt {7}+4 x +1\right )^{\frac {3}{4}} \left (i \sqrt {7}+4 x +1\right )^{-\frac {3 i \sqrt {7}}{28}} \left (i \sqrt {7}-4 x -1\right )^{\frac {3}{4}} \left (i \sqrt {7}-4 x -1\right )^{\frac {3 i \sqrt {7}}{28}} y}{1+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {1}{2}, 2\right ], \left [\frac {\left (7 \sqrt {7}-3 i\right ) \sqrt {7}}{28}\right ], \frac {1}{2}+\frac {i \left (-4 x -1\right ) \sqrt {7}}{14}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [\frac {1}{2}, 2\right ], \left [\frac {\left (7 \sqrt {7}-3 i\right ) \sqrt {7}}{28}\right ], \frac {1}{2}+\frac {i \left (-4 x -1\right ) \sqrt {7}}{14}\right )}\right ] \\ \end{align*}