Link to actual problem [15375] \[ \boxed {y^{\prime \prime \prime \prime }-y=8 \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = 4, y^{\prime \prime \prime }\left (0\right ) = 6] \end {align*}
type detected by program
{"higher_order_linear_constant_coefficients_ODE"}
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*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {y}{2}+x \,{\mathrm e}^{x}\right ] \\ \\ \end{align*}