Link to actual problem [9838] \[ \boxed {x^{3} y^{\prime \prime \prime }+6 y^{\prime \prime } x^{2}+\left (a \,x^{3}-12\right ) y=0} \]
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 &= \frac {\left (x \,a^{3}+2 \left (-a^{4}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} x}{a}}}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} x}{a}} y}{x \,a^{3}+2 \left (-a^{4}\right )^{{2}/{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (i x \,a^{3}+\left (-a^{4}\right )^{{2}/{3}} \sqrt {3}-i \left (-a^{4}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2 a}} x^{3} y}{\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{{2}/{3}}+i x \,a^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2 a}} \left (-i x \,a^{3}+\left (-a^{4}\right )^{{2}/{3}} \sqrt {3}+i \left (-a^{4}\right )^{{2}/{3}}\right )}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} {\mathrm e}^{\frac {\left (-a^{4}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2 a}} y}{\left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{{2}/{3}}-i x \,a^{3}}\right ] \\ \end{align*}