1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans! orm Method

5.4 The Unit Function and

Time-Delay Function

257

The Laplace transform method is useful not only for solving nonhomo-
geneous linear differential equations and initial value problems with con-
stant coefficients when the nonhomogeneity- which is also called the forc-
ing function- is a solution of some homogeneous linear differential equation
with constant coefficients but also when the forcing function is a discontinuous
function or an impulse function which h as a Laplace transform. Forcing func-
tions which are discontinuous functions or impulse functions occur frequently
in electrical and mechanical systems. In this section, we shall consider forcing
functions which are discontinuous functions and in the next section we shall
consider forcing functions which are impulses.

One of the simplest functions which has a jump discontinuity of 1 at

x = c ~ 0 is the unit step function or the Heaviside function, u(x - c),

which is named in honor of Oliver Heaviside (1850-1925).

DEFINITION The Unit Step Function or Heaviside Function

The unit step function or Heaviside function is

{

o,

u(x - c) =

1,

x<c

c::;x

Graphs of u(x - 1), 2u(x - 1), and -u(x - 2) are displayed in Figure 5.3.
Other discontinuous step functions can be written as a linear combination of
unit step functions. For instance,

(1)

l

0, x < 1

h(x)= 2, l::;x<2

1, 2::;x

can be written as h(x) = 2u(x - 1) - u(x - 2). Thus, the unit step function
is the basic function for constructing other step functions. A graph of h(x) is
shown in Figure 5.3d.