What I'm given: $$x'' = x' + y' + x + y$$ $$y'' = 2x' + 3y' + 3x + y$$ $$z=x'$$ $$w=y'$$
My solution: We know that $z'=x''$ and $w'=y''$. We can write:
$$z'=z+w+x+y$$ $$w'=2z+3w+3x+y$$
I'm not entirely sure what I'm supposed to do with the $x$ and $y$ still stuck in the derived first order equations, since I'm given $z=x'$, $w=y'$, and I'm only supposed to get a total of four first-order equations (so I can't define, for example, $z_0=∫zdx=x$). Any help is greatly appreciated!
HINT :
$ \left(\begin{matrix} x' \\ y' \\ z' \\ w' \\ \end{matrix}\right)= \left(\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ 3 & 1 & 2 & 3 \\ \end{matrix}\right) \left(\begin{matrix} x \\ y \\ z \\ w \\ \end{matrix}\right)$
See an example of solving in : https://en.wikipedia.org/wiki/Matrix_exponential#Linear_differential_equation_systems in section "Example (homogeneous)".