Link to actual problem [9777] \[ \boxed {y^{\prime \prime \prime }-a \,x^{b} y=0} \]
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 [\right ], \left [\frac {b +1}{b +3}, \frac {b +2}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +1}{b +3}, \frac {b +2}{b +3}\right ], \frac {a \,x^{b} x^{3}}{\left (b +3\right )^{3}}\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 [\right ], \left [\frac {b +2}{b +3}, \frac {b +4}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +2}{b +3}, \frac {b +4}{b +3}\right ], \frac {a \,x^{b} x^{3}}{\left (b +3\right )^{3}}\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 [\right ], \left [\frac {b +4}{b +3}, \frac {b +5}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +4}{b +3}, \frac {b +5}{b +3}\right ], \frac {a \,x^{b} x^{3}}{\left (b +3\right )^{3}}\right )}\right ] \\ \end{align*}