Link to actual problem [2516] \[ \boxed {f^{\prime \prime }+6 f^{\prime }+9 f={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [f \left (0\right ) = 0, f^{\prime }\left (0\right ) = \lambda ] \end {align*}
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 &= t, S \left (R \right ) &= \frac {{\mathrm e}^{3 t} f}{t}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}