Link to actual problem [14624] \[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+y={\mathrm e}^{-\frac {t}{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = a, y^{\prime }\left (0\right ) = b] \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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {y}{2}\right ] \\ \left [R &= y \,{\mathrm e}^{\frac {t}{2}}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y \,{\mathrm e}^{\frac {t}{2}}}{t^{2}+8 t +16}, S \left (R \right ) &= -\frac {\ln \left (-t -4\right )}{2}\right ] \\ \end{align*}