Link to actual problem [9896] \[ \boxed {y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (2 x \right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y}{\sin \left (2 x \right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (2 x \right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y}{\sin \left (2 x \right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )}\right ] \\ \end{align*}