Link to actual problem [7190] \[ \boxed {y^{\prime \prime \prime }-y^{\prime } x^{3}-y x^{2}=x^{3}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {1}{5}\right ], \left [\frac {3}{5}, \frac {4}{5}\right ], \frac {x^{5}}{25}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [\frac {1}{5}\right ], \left [\frac {3}{5}, \frac {4}{5}\right ], \frac {x^{5}}{25}\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 {2}{5}\right ], \left [\frac {4}{5}, \frac {6}{5}\right ], \frac {x^{5}}{25}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [\frac {2}{5}\right ], \left [\frac {4}{5}, \frac {6}{5}\right ], \frac {x^{5}}{25}\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 {3}{5}\right ], \left [\frac {6}{5}, \frac {7}{5}\right ], \frac {x^{5}}{25}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {hypergeom}\left (\left [\frac {3}{5}\right ], \left [\frac {6}{5}, \frac {7}{5}\right ], \frac {x^{5}}{25}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x}{2}+y\right ] \\ \\ \end{align*}