Link to actual problem [13175] \[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y={\mathrm e}^{-\frac {t}{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
type detected by program
{"kovacic", "second_order_linear_constant_coeff"}
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 [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{2 t} y}{\sin \left (4 t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{2 t} y}{\cos \left (4 t \right )}\right ] \\ \end{align*}
\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*}