Phase plane interpretations I'm really quite confused about what phase planes/portraits represent. I'm only considering homogeneous systems. 
Suppose we have the system,
$$ \begin{bmatrix}x_1'  \\x_2'  \end{bmatrix} = \begin{bmatrix}1 & 2 \\3 & 2 \end{bmatrix}\begin{bmatrix}x_1  \\x_2  \end{bmatrix}$$
The associated phase portrait is then, 

I understand that to get the picture above, we had to take a number of points, multiply them by the coefficient matrix and plot the resulting points in the $x_1 - x_2$ plane. From what I've been told, the resulting phase plane is a graphical representation of $\frac{dx_1 }{dx_2 }$. That is what really confuses me. Why is that of any interest at all? How is that meant to give us the 'trajectories' of our solution when it doesn't contain '$t$' (assuming $x_1$ and $x_2$ are functions of $t$)
 A: You can draw the actual solution curves $x_1=x_1(t)$, $x_2=x_2(t)$ in a three-dimensional coordinate system $(t,x_1,x_2)$. The phase portrait is what you get if you project this picture onto the $(x_1,x_2)$ plane.
A: It's an interpretation of the solutions of the equation as representing parametrisations of curves in a plane with a coordinate system $(x_1,x_2)$. Something like knowing the possible trajectories/orbits of a planet are elliptical, but not knowing on what ellipse (which requires an initial condition), and where at what time the planet is on a certain ellipse (which requires to know the $t$ value associated with the points $(x_1,x_2)$.
The reason one might be interested in these phase portraits, is that you can at a glance see features of your solutions. For instance, by inspecting your picture, I can see that the point $(0,0)$ is repulsive for your trajectories. A solution will never go through it. Solutions tend to start from a point far away from $(0,0)$, come close to it and then bounce away. Also, you can divide the plane in 4 regions. With an initial condition in one of those regions, you can never reach a different region.
Take a look at these notes to see the kind of things you can read off of a phase portrait.
