1550078481-Ordinary_Differential_Equations__Roberts_

258 Ordinary Differential Equations

a. y(x) u(x - 1)

-4 -2

T

x

2 4

b. y(x) 2u(x-1)

-4 -2

T

x

2 4

c. y(x) -u(x - 2)

T

x

-2 -4 2._ __^4

d. y(x) 2 u(x - 1) -u(x - 2)

-4 -2

T

x

2 4

Figure 5.3 Graphs of Step Functions

We easily calculate the Laplace transform of the unit step function as fol-

lows:

1

00

.C[u(x - c)J = u(x - c)e-sx dx = 1= e-sx dx = e-cx -- for s > 0.

0 c s

Using the linearity property of the Laplace transforms, we will be able to

calculate the Laplace transform of any particular step function once it is

written as a linear combination of unit step functions. For example, the

Laplace transform of h(x) = 2u(x - 1) - u(x - 2) is calculated as follows:

2e-s e-2s

.C[h(x) ] = .C[2u(x -1) -u(x - 2)] = 2.C[u(x -1)] -.C[u(x - 2)] = - - -

s s

for s > 0.