# Forced Equations

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

Principle.

**Extended Linearity Principle** Consider a
nonhomogeneous equation (a forced equa-

tion)

and its corresponding homogeneous equation (the unforced equation)

1. Suppose is a
particular solution of the nonhomogeneous equation and
is

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
somewhat differently.

**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