Chapter 6
Applications of Linear Differential
Equations with Constant Coefficients
In this chapter we will examine a few linear differential equations with con-
stant coefficients which arise in the study of various physical and electrical
systems. Usually the independent variable, t, will represent time and the de-
pendent variable or solution, y(t), will represent a parameter which describes
the state of the system.6.1 Second-Order Differential Equations
Second-order linear differential equations provide mathematical models for
various physical phenomena. As a matter of fact, several physical phenomena
often give rise to the same mathematical model. Therefore, it is sometimes
possible to simulate the behavior of an expensive physical system such as an
airplane, automobile, or bridge, etc., by another inexpensive physical system
such as an electrical circuit. In this section, we present the mathematical
models which arise from a simple pendulum, a mass on a spring, and an R LC
electrical circuit and discover that they are all the same model. This leads us
to examine in detail the second order linear differential equation with constant
coefficients.
A Simple P endulum A simple pendulum consists of a rigid, straight rod
of negligible mass and length e with a bob of mass m attached to one end. The
other end of the rod is attached to a fixed support so that the pendulum is
free to move in a vertical plane. Let y denote the angle, expressed in radians,
which the rod makes with the vertical- the equilibrium position of the system.
We arbitrarily choose y to be positive if the rod is to the right of vertical and
negative if the rod is to the left of vertical as shown in Figure 6.1.
We will assume the only forces acting on the pendulum are the force of
gravity and a force due to air resistance which is proportional to the velocity
of the bob. Under these assumptions it can be shown by applying Newton's
second law of motion that the position of the pendulum satisfies the initial
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