Link to actual problem [9857] \[ \boxed {y^{\prime \prime \prime }-y^{\prime } x -n y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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 &= \operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{9}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {hypergeom}\left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right )}\right ] \\ \end{align*}