Link to actual problem [9682] \[ \boxed {y^{\prime \prime }+\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}-\frac {x y}{x^{3}+1}=0} \]
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 (x^{3}+1\right )^{\frac {1}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x^{3}+1\right )^{\frac {1}{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {4}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}\right ] \\ \end{align*}