Link to actual problem [7553] \[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x=0} \]
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}{\left (x^{3}+1\right )^{\frac {4}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{3}+1\right )^{\frac {4}{3}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [1, 1\right ], \left [\frac {2}{3}\right ], -x^{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [1, 1\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}\right ] \\ \end{align*}