Link to actual problem [12314] \[ \boxed {y^{\prime \prime }+2 y^{\prime }+y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
type detected by program
{"second_order_laplace", "second_order_linear_constant_coeff", "linear_second_order_ode_solved_by_an_integrating_factor"}
type detected by Maple
[[_2nd_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 [R &= \frac {y \,{\mathrm e}^{t}}{t^{2}+4 t +4}, S \left (R \right ) &= -\ln \left (-2-t \right )\right ] \\ \end{align*}