Link to actual problem [9911] \[ \boxed {x^{{5}/{2}} y^{\left (5\right )}-y a=0} \]
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 &= -8 x \,a^{{2}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-4 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} x \,a^{{2}/{5}}+12 \sqrt {x}\, {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-12 \sqrt {x}\, {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-6 \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+6 \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+6 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} \sqrt {x}\, a^{{1}/{5}}-3 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} y}{12 \sqrt {x}\, {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-12 \sqrt {x}\, a^{{1}/{5}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-6 \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-8 \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) x \,a^{{2}/{5}}-8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+\left (-4 x \,a^{{2}/{5}}+6 \sqrt {x}\, a^{{1}/{5}}-3\right ) {\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 8 x \,{\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-8 x \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-4 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} x \,a^{{2}/{5}}+12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-6 \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+6 \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+6 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} \sqrt {x}\, a^{{1}/{5}}-3 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} y}{-6 \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+8 x \,a^{{2}/{5}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-8 x \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+12 \sqrt {x}\, a^{{1}/{5}} \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+\left (-4 x \,a^{{2}/{5}}+6 \sqrt {x}\, a^{{1}/{5}}-3\right ) {\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 4 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} x \,a^{{2}/{5}}-6 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} \sqrt {x}\, a^{{1}/{5}}+3 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}}+8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-8 x \,a^{{2}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \,{\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-6 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} y}{6 \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-6 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-8 \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) x \,a^{{2}/{5}}+12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-12 \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) \sqrt {x}\, a^{{1}/{5}}+\left (-6 \sqrt {x}\, a^{{1}/{5}}+4 x \,a^{{2}/{5}}+3\right ) {\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -8 x \,{\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+8 x \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-4 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} x \,a^{{2}/{5}}+12 \sqrt {x}\, {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-12 \sqrt {x}\, {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \,{\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-6 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} \sqrt {x}\, a^{{1}/{5}}-3 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} y}{12 \sqrt {x}\, a^{{1}/{5}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-12 \sqrt {x}\, {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{1}/{5}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+8 x \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} a^{{2}/{5}} \cos \left (\frac {\pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-8 x \,a^{{2}/{5}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )-6 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+\left (-4 x \,a^{{2}/{5}}+6 \sqrt {x}\, a^{{1}/{5}}-3\right ) {\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} \sqrt {x}\, a^{{1}/{5}}}{5}+\frac {4 \,{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} x \,a^{{2}/{5}}}{15}+\frac {{\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}}}{5}+\frac {4 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{5}+\frac {8 x \,a^{{2}/{5}} {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{15}+\frac {2 \,{\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{5}-\frac {8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{15}-\frac {4 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{5}+\frac {2 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )}{5}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {15 \,{\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} y}{8 \sin \left (\frac {\pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) x \,a^{{2}/{5}}+12 \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) \sqrt {x}\, a^{{1}/{5}}-8 x \,a^{{2}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \sin \left (\frac {3 \pi }{10}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}-12 \sqrt {x}\, a^{{1}/{5}} {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (\frac {2 \pi }{5}+2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+6 \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right )+6 \,{\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}} \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}\right ) {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}+\left (-6 \sqrt {x}\, a^{{1}/{5}}+4 x \,a^{{2}/{5}}+3\right ) {\mathrm e}^{2 \sqrt {x}\, a^{{1}/{5}}} {\mathrm e}^{2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{{1}/{5}}}}\right ] \\ \end{align*}