Link to actual problem [9795] \[ \boxed {y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y=0} \]
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 &= {\mathrm e}^{-\frac {x^{2} a}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= y \,{\mathrm e}^{\frac {x^{2} a}{2}}\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 {3}\, \sqrt {a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2} a}{2}} {\mathrm e}^{-\sqrt {3}\, \sqrt {a}\, 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 {3}\, \sqrt {a}\, x}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2} a}{2}} {\mathrm e}^{\sqrt {3}\, \sqrt {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*}