Link to actual problem [12032] \[ \boxed {x^{\prime \prime }+2 x^{\prime }+x={\mathrm e}^{-t}} \]
type detected by program
{"kovacic", "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 {x \,{\mathrm e}^{t}}{t^{2}+4 t +4}, S \left (R \right ) &= -\ln \left (-2-t \right )\right ] \\ \end{align*}