Link to actual problem [9849] \[ \boxed {x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+y a=0} \]
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 &= \frac {\left (-8 x^{3}+a \right )^{4} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}}}{a x}}}{{\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{3} \left (4 a^{2} x^{2}-2 \left (-a^{4}\right )^{{1}/{3}} x a +\left (-a^{4}\right )^{{2}/{3}}\right )^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}}}{a x}} {\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{3} \left (4 a^{2} x^{2}-2 \left (-a^{4}\right )^{{1}/{3}} x a +\left (-a^{4}\right )^{{2}/{3}}\right )^{4} y}{\left (-8 x^{3}+a \right )^{4}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (-8 x^{3}+a \right )^{4} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a x}}}{{\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 i a x +i \left (-a^{4}\right )^{{1}/{3}}\right )}^{3} {\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 i a x -i \left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a x}} {\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 i a x +i \left (-a^{4}\right )^{{1}/{3}}\right )}^{3} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} y}{\left (-8 x^{3}+a \right )^{4}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (-8 x^{3}+a \right )^{4} {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a x}}}{{\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 i a x -i \left (-a^{4}\right )^{{1}/{3}}\right )}^{3} {\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}-4 i a x +i \left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a x}} {\left (\left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 i a x -i \left (-a^{4}\right )^{{1}/{3}}\right )}^{3} {\left (i \left (-a^{4}\right )^{{1}/{3}} \sqrt {3}+4 x a -\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} {\left (2 x a +\left (-a^{4}\right )^{{1}/{3}}\right )}^{4} y}{\left (-8 x^{3}+a \right )^{4}}\right ] \\ \end{align*}