Link to actual problem [7582] \[ \boxed {2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (x +1\right ) y=0} \]
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 {x^{\frac {1}{3}}}{\left (x +3\right ) \left (1+\frac {x}{3}\right )^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +3\right ) \left (1+\frac {x}{3}\right )^{\frac {1}{3}} y}{x^{\frac {1}{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x}{3}\right )}\right ] \\ \end{align*}