

Appendix: SecondOrder Differential Equations as Systems
Every secondorder differential equation may be considered as a system of two firstorder equations. This has a number of uses. In particular, it enables us to use firstorder numerical methods to approximate the solution of a secondorder initial value problem.In order to have a particular example, let's consider the secondorder 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 firstorder 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 d^{2}/dt^{2}. So our new system of equations becomes
with initial conditions u(0) = _{0} and v(0) = v_{0}.
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), u_{0} = 3, and v_{0} = 0 (zero initial velocity). We use numerical methods to plot the direction field and the trajectory in the phase plane.
Note that there are many ways of defining the new dependent variables u and v when creating a system corresponding to the original secondorder 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.
(b) For each of the points A  F on the trajectory, describe the motion of the pendulum at the corresponding time.
Describe the pendulum motion corresponding to Pendulum Trajectory 2.
Describe the pendulum motion corresponding to Pendulum Trajectory 3.


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