Link to actual problem [9832] \[ \boxed {x^{3} y^{\prime \prime \prime }+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y=0} \]
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 &= x \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\nu }{3}, 1+\frac {\nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\nu }{3}, 1+\frac {\nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\nu +1} \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\nu }{3}, 1-\frac {2 \nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\nu } y}{x \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\nu }{3}, 1-\frac {2 \nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\nu +1} \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {\nu }{3}, 1+\frac {2 \nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\nu } y}{x \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {\nu }{3}, 1+\frac {2 \nu }{3}\right ], -\frac {a \,x^{3}}{27}\right )}\right ] \\ \end{align*}