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Systems of Differential Equations: Models of Species Interaction

Appendix: Second-Order Differential Equations as Systems

Every second-order differential equation may be considered as a system of two first-order equations. This has a number of uses. In particular, it enables us to use first-order numerical methods to approximate the solution of a second-order initial value problem.

In order to have a particular example, let's consider the second-order equation describing the motion of of a pendulum.

Here is the angle the pendulum has moved from the vertical, L is the length of the pendulum, g is the acceleration due to gravity, m is the mass of the pendulum, and b is a damping coefficient. See the figure below.

The statement itself is a consequence of Newton's Second Law of Motion. To see more detail on the derivation click here. However, for our immediate purposes all we need is the differential equation itself.

So suppose we have an initial value problem consisting of the pendulum differential equation together with initial conditions

We will construct a system of first-order equations in the dependent variables u and v and assign initial values so that the new initial value problem is equivalent to the original problem. First we let u = and v = d/dt. Then dv/DT is the same as d2/dt2. So our new system of equations becomes

with initial conditions u(0) = 0 and v(0) = v0.

Now we can apply our numerical methods for a system to describe the solution of this problem. This is important since the presence of the sine in the equation makes it impossible for us to obtain a symbolic description of the solution.

Let's look at an example. We'll use units of meters and seconds, so we may take g = 9.807. Assume m = 1, and L = 1. We'll begin with b = 0 (no damping), u0 = -3, and v0 = 0 (zero initial velocity). We use numerical methods to plot the direction field and the trajectory in the phase plane.

Pendulum Trajectory 1

Note that there are many ways of defining the new dependent variables u and v when creating a system corresponding to the original second-order differential equation. One advantage to the approach we have used is that the phase plane plots the dependent variable (in our case u = ) on the horizontal axis and its derivative (in our case v = d/DT) on the vertical axis. This enables us to give a straightforward interpretation to the coordinates of points on a trajectory.

    1. (a) In what direction is the trajectory being traced out?

      (b) For each of the points A - F on the trajectory, describe the motion of the pendulum at the corresponding time.

  2. Here is the trajectory corresponding to the same coefficients as in Step 1, but now with initial conditions u0 = -3, and v0 = 5.

    Pendulum Trajectory 2

    Describe the pendulum motion corresponding to Pendulum Trajectory 2.

  3. Now we change b to 0.2 (and leave the other coefficients alone). We use the same initial conditions.

    Pendulum Trajectory 3

    Describe the pendulum motion corresponding to Pendulum Trajectory 3.

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