For the last five weeks, all of our differential equations
have been autonomous. Now we
turn to second-order equations that model systems that are subject to some type of external
forcing. Here are two examples:
Example. The nonlinear pendulum with a pivot point that is subject to vertical oscillations.
The motion of such a pendulum is governed by the second-order nonlinear equation
where ω determines the frequency of the oscillations of
the pivot point and F determines
the amplitude of the oscillations. The Pendulums tool on the CD illustrates this system.
Example. The linear mass-spring system where the
spring is subject to vertical oscilla-
tions. To model this system, we use the standard mass-spring system and add a term that
corresponds to the force added to the system by the oscillations. We get
The standard ForcedMassSpring tool on the CD illustrates this system.
In class we will discuss forced linear equations only, but
your second project will involve
some experimentation with the forced pendulum.
Our success studying unforced linear systems was due in
large part to the Linearity
Principle. For forced linear equations, we are fortunate to have the Extended Linearity
Extended Linearity Principle Consider a
nonhomogeneous equation (a forced equa-
and its corresponding homogeneous equation (the unforced equation)
1. Suppose is a
particular solution of the nonhomogeneous equation and
a solution of the corresponding homogeneous equation. Then is also a
solution of the nonhomogeneous equation.
2. Suppose and are two solutions of the nonhomogeneous equation. Then
is a solution of the corresponding homogeneous equation.
Therefore, if is the general solution of the homogeneous equation, then
is the general solution of the nonhomogeneous equation.
This principle provides the basic framework that we will use to solve linear second-order
forced equations. (At this point in the course, you should go back and review the method
described in Section 1.8 for solving nonhomogeneous first-order linear equations.)
We already know how to find the general solution to the
associated homogeneous equa-
tion, so we need only find one solution to the original equation.
Example 1. Consider the equation
Here’s another example that looks similar but goes
Example 2. Consider the equation
A time saver: There’s a calculation that we’ve already
done twice before. It is also useful
for guessing . Consider the function and calculate
Let’s see how this works in Example 1.
Example 1. Recall