Link to actual problem [9885] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_high_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 [R &= x, S \left (R \right ) &= \frac {y}{x \left (\operatorname {KelvinBer}\left (n , x\right )^{2}+\operatorname {KelvinBei}\left (n , x\right )^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \left (\operatorname {KelvinBer}\left (-n , x\right )^{2}+\operatorname {KelvinBei}\left (-n , x\right )^{2}\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 {3}{2}, \frac {n}{2}+1, -\frac {n}{2}+1\right ], \frac {x^{4}}{64}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{2}, \frac {n}{2}+1, -\frac {n}{2}+1\right ], \frac {x^{4}}{64}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {1}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \frac {x^{4}}{64}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {1}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \frac {x^{4}}{64}\right )}\right ] \\ \end{align*}