Link to actual problem [9869] \[ \boxed {y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y=0} \]
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 &= {\mathrm e}^{-\frac {x^{2} a}{2}+\sqrt {\left (3-\sqrt {6}\right ) a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\sqrt {-a \left (-3+\sqrt {6}\right )}\, x} {\mathrm e}^{\frac {x^{2} a}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2} a}{2}+\sqrt {\left (3+\sqrt {6}\right ) a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\sqrt {\left (3+\sqrt {6}\right ) a}\, x} {\mathrm e}^{\frac {x^{2} a}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2} a}{2}-\sqrt {\left (3-\sqrt {6}\right ) a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2} a}{2}} {\mathrm e}^{\sqrt {-a \left (-3+\sqrt {6}\right )}\, x} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2} a}{2}-\sqrt {\left (3+\sqrt {6}\right ) a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2} a}{2}} {\mathrm e}^{\sqrt {\left (3+\sqrt {6}\right ) a}\, x} y\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y \,{\mathrm e}^{\frac {x^{2} a}{2}}, S \left (R \right ) &= x\right ] \\ \end{align*}