Link to actual problem [6664] \[ \boxed {2 y^{\prime \prime \prime }+3 y^{\prime \prime }-3 y^{\prime }-2 y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
type detected by program
{"higher_order_laplace"}
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -y +\frac {{\mathrm e}^{-t}}{2}\right ] \\ \\ \end{align*}