# Linear System: Phase Planes

• March 22nd 2009, 11:51 PM
utopiaNow
Linear System: Phase Planes
Hi Everyone,

The system given is:

$

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

$

The eigenvalues and the corresponding eigenvectors I found were:

$
r_1 = 2, r_2 = 4

$

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

Therefore,

$
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 $x_1$ 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.
• March 23rd 2009, 07:30 AM
HallsofIvy
Quote:

Originally Posted by utopiaNow
Hi Everyone,

The system given is:

$

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

$

The eigenvalues and the corresponding eigenvectors I found were:

$
r_1 = 2, r_2 = 4

$

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

Therefore,

$
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 $x_1$ 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.

First graph the two lines given by the eigenvectors: <1, 3> lies along the line y= 3x and <1, 1> along the line y= x. Since both eigenvalues are positive, the "flow" is outward. All other solutions will be outward along curves emanating from the origin curving toward y= x because it corresponds to the larger eigenvalue, 4 so the flow is "faster" there.