Link to actual problem [9822] \[ \boxed {x^{2} y^{\prime \prime \prime }+6 y^{\prime \prime } x +6 y^{\prime }+a \,x^{2} y=0} \]
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 {{\mathrm e}^{\left (-a \right )^{{1}/{3}} x}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} {\mathrm e}^{-\left (-a \right )^{{1}/{3}} x} y\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 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2}}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} {\mathrm e}^{-\frac {\left (-a \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) x}{2}} y\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 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2}}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{2} {\mathrm e}^{\frac {\left (-a \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) x}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 y}{x}\right ] \\ \\ \end{align*}