How to write a system of equations as a dynamical system? I am having a lot of trouble understanding how to move from a system of ODE's to a dynamical systems point of view (that will allow me to make a phase-plane analysis). 
Assume I want to write the following system (I invented it just as an example):
$$
\begin{split}
y''+2x+3y'+4x'=0\\
2y''+2x-4y'+2x'=0
\end{split}
$$
in a matrix form:
$$
\frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = A\begin{bmatrix} x\\x'\\y\\y' \end{bmatrix}
$$
where $A$ is a $4\times 4$ matrix. 
I know I can write this system as:
$$
\begin{bmatrix} 0&0&0&1\\0&0&0&2\end{bmatrix} \frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = \begin{bmatrix} -2&4&0&-3\\-2&-2&0&4\end{bmatrix}  \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix}
$$
but it does not help me much (the inverse of the matrix in the LHS is not a well defined notion...).
In addition, I know that if I would only had the first equation, I would be able to write it as:
$$
\frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = \begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&1\\-2&-4&0&-3 \end{bmatrix}\begin{bmatrix} x\\x'\\y\\y' \end{bmatrix}
$$
but how can I combine the two equations into such a form (in order to make a phase-plane analysis) ??
Just to clarify- I know that one possible solution is to isolate $y''$ from the first equation, substitute in the second one, and then move to a matrix form, but I don't want to do it (too messy and not very helpful in case of a lot of variables and a lot of equations).
Will you please help me?
Thanks 
 A: Note that $\det \mathbf{A}=0$ and both $x'$ and $y'$ appear on both sides.  How about in this way?
$$\left(
   \begin{array}{c}
     x' \\ y'' \end{array}
  \right)=
 \left(
   \begin{array}{cc}
     -\frac{1}{3} & -\frac{5}{3} \\
     -\frac{2}{3} & \frac{11}{3}
   \end{array}
 \right)
 \left(
   \begin{array}{c}
     x \\ y'
   \end{array}
 \right)$$
A: Strictly speaking, if you decide just to write down two linear equations involving $x$, $y$ and their (first and second) time derivatives, there is no a priori reason to assume those two equations are consistent with each other. Therefore, the best thing to do is to write each equation separately as a four-dimensional linear system of the form $\mathbf{x}' = A \mathbf{x}$, and solve those two. If a solution obeys both original equations, it has to solve both dynamical systems simultaneously. By comparing their general solutions, you can then see if these overlap in some way.
Note that I didn't go into details regarding the example you gave, since I've got the impression your question is not about this example in particular, but about the general approach.
