Link to actual problem [12310] \[ \boxed {4 y^{\prime \prime }+5 y^{\prime }+4 y=3 \,{\mathrm e}^{-t}} \] 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"}
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {5 t}{8}} \sin \left (\frac {\sqrt {39}\, t}{8}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 t}{8}} y}{\sin \left (\frac {\sqrt {39}\, t}{8}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {5 t}{8}} \cos \left (\frac {\sqrt {39}\, t}{8}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 t}{8}} y}{\cos \left (\frac {\sqrt {39}\, t}{8}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}