Link to actual problem [13224] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+5 y=2 \,{\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, 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 [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-2 t} y}{\sin \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-2 t} y}{\cos \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}