Math 240 Home
Launch Systems Lab
Behavior of Solutions to First-Order Linear Systems
The solution of a system is a pair of functions, x(t) and
y(t) (or even three or more functions depending on the size
of the system). To view the solution, it is usually most helpful to
consider (x(t), y(t)) as a parametric
curve which can be plotted on the xy-plane. In this lab we will
look at a variety of such curves and develop some ideas about how the
pictures relate to the initial equation.
- Launch the Systems lab (use the link at the top of the page). You can
now enter a constant-coefficient first-order system for x and
y, along with a pair of initial values. If you press the
"Update Equation" button, the graph will draw a pair of lines. The
magenta (light purple) line marks where x' = 0 and the
cyan (light blue) line marks where y' = 0. The arrows
show the direction the parametric curve will travel when it crosses the
line (as t increases). The button marked "Draw Integral
Curve" will draw the parametric curve for the solution with the
listed initial values (note that the curve will start at initial point
listed). If you get too many curves on the screen, you can press the
"Update Equation" button to clear off all the curves.
- Experiment with different systems and initial values. Note that the
solution curves are always horizontal when they cross the cyan line and
vertical when they cross the magenta line. Why do the curves behave
- Now find examples of systems of each of the following 6 types of
that you don't have to find examples that look exactly like the pictures,
just examples that that have the same sort of behavior. Of course, using
the magenta and cyan lines you can actually match the pictures without
much effort, and that is probably the easiest way to find examples of some
of the behaviors.
Solutions diverge straight out to infinity
Solutions converge directly to the origin (possibly with a "twist" as
A "saddle point" at (0,0) where solutions converge toward the origin
in one direction and then turn and diverge away in a different direction
(there will be one solution converging to the origin in between the
solutions turning in each direction).
Solutions spiral out to infinity.
Solutions spiral in to the origin.
Solutions spin around the origin along ellipses, neither converging
toward the origin nor diverging away from it.
- In the system
x' = ax + by,
y' = cx + dy,
two very useful quantities to know are the
trace = a + d and the
determinant = ad - bc. Compute the trace
and determinant for each of your examples. Can you find a relationship
between the type of system (saddle, spiral in, etc.) and the values of the
trace and determinant? Many of these are straightforward, but
distinguishing between a couple of cases will be a little subtle.
You will want to check some other examples to be sure you have the
- Finally, write the Laplace transform of the solutions to the system
x' = ax + by,
y' = cx + dy,
in terms of the coefficients, a, b, c, d,
x0, and y0.
Identify where the trace and determinant appear in the formulas for the
solution. Explain why knowledge of the trace and determinant should enable
you to predict the form of the solution. You don't have to justify
your 6 relationships, just relate what you discover to a previous lab to
explain in general why knowledge of the trace and determinant can give you
so much information about the shape of the solution.
Prepare a lab report for this lab which includes your results for the
questions/instructions in bold face.
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©2002 Andrew G. Bennett