I'm stuck on that problem, none of the examples provided match this type of question. How am I supposed to proceed ?
Solve the following system of ODE
\begin{align*}
\ddot x + 3\dot x - 2x + \dot y - 3y & = 2e^{-t} \\
2\dot x - x +\dot y - 2y & = 0
\end{align*}
with initial conditions $x(0) = \dot x (0) = 0$ and $y(0) = 4$.
The usual way is to convert the 2nd-order equations into 1st-order equations. Let $z=x'$ and $w=y'.$ Then your first equation becomes
$$z'= -3z +2x -w +3y +2e^{-t}$$
and your second equation becomes
$$y'=-2z+x+2y$$
and since $y'=w$, you have $w=-2z+x+2y.$
Now you have the system of equations
$$x' = z$$ $$y' = -2z+x+2y$$ $$z' = -3z+2x-w+3y +2e^{-t}$$
Plug in the above expression for $w$ and you have a first-order system in 3 variables. Note that $z(0) = x'(0) = 0.$ Proceed by covering up the $2e^{-t}$ and solving the homogeneous system first. Then follow with variation of parameters (or undetermined coefficients) to solve the non-homogeneous version.