I know how to determine the type and stability of the equilibrium point using eigenvalues and eigenvectors. But if I have a system lets say
$$\dot{x}=x, \,\,\, \dot{y}=y$$
we can see that $x(t)$ is increasing and $y(t)$ is also increasing. Does that mean that the trajectories are all going towards the increasing $x$ and $y$ directions.
I imagine to solve these equations I use the chain rule and I'll get a separable differential equation? Then that would give me the general equation.
Does this phase plane seem correct to you?
In this case the solutions are
\begin{eqnarray} x(t) &=& x_0e^{t} \\ y(t) &=& y_0e^{t} \end{eqnarray}
which can be written as
\begin{eqnarray} y_0 x(t) &=& y_0 x_0 e^{t} \\ x_0 y(t) &=& y_0 x_0 e^{t} \end{eqnarray}
Therefore
$$ x_0 y = y_0 x $$
which are straight lines going out from the center