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.


  1. 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.

  2. 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 like that?

  3. Now find examples of systems of each of the following 6 types of systems. Note 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.
    (A) Solutions diverge straight out to infinity
    (B) Solutions converge directly to the origin (possibly with a "twist" as illustrated here).
    (C) 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).
    (D) Solutions spiral out to infinity.
    (E) Solutions spiral in to the origin.
    (F) Solutions spin around the origin along ellipses, neither converging toward the origin nor diverging away from it.

  4. In the system

    x' = ax + by, x(0)=x0,
    y' = cx + dy, y(0)=y0,

    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 relationship correct.

  5. Finally, write the Laplace transform of the solutions to the system

    x' = ax + by, x(0)=x0,
    y' = cx + dy, y(0)=y0,

    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 each of 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