Link to actual problem [7562] \[ \boxed {\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (x +1\right ) y^{\prime }+8 y=0} \]
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 {2+x}{\left (1+\frac {2 x}{3}\right )^{\frac {2}{3}} x^{\frac {7}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+\frac {2 x}{3}\right )^{\frac {2}{3}} x^{\frac {7}{3}} y}{2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [2, 2\right ], \left [\frac {10}{3}\right ], -\frac {2 x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [2, 2\right ], \left [\frac {10}{3}\right ], -\frac {2 x}{3}\right )}\right ] \\ \end{align*}