Linear System: Phase Planes

Hi Everyone,

The system given is:

$\displaystyle

\frac{dX}{dt} = \left(\begin{array}{cc}5&-1\\3&1\end{array}\right)X

$

The eigenvalues and the corresponding eigenvectors I found were:

$\displaystyle

r_1 = 2, r_2 = 4

$

$\displaystyle

\xi_1 = (1, 3)^T, \xi_2 = (1, 1)^T

$

Therefore,

$\displaystyle

x = c_1\left(\begin{array}{c}1\\3\end{array}\right)e^{ 2t} + c_2\left(\begin{array}{c}1\\2\end{array}\right)e^{ 4t}

$

Now the text says to sketch several trajectories in the phase plane and also sketch some typical graphs of $\displaystyle x_1$ versus t .

The text does not really explain how its generating the example graphs, it just shows pictures of general graphs. How would I go about sketching these? Without using a computer of course.