1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans/ arm Met hod 261

x c

a. b.

x

Figure 5.6 The Function f(x) and Associated c-time Delay Function g(x)

The following theorem provides us with the relationship between the Laplace
transform of a function f(x) and the Laplace transform of the c-time delay
function of f(x), u(x - c)f(x - c).

THEOREM 5.3 SECOND TRANSLATION PROPERTY

If £[/(x) ] exists for s >a:?'. 0 and if c > 0, then

.C[u(x - c)J(x - c)] = e-cs .C[f(x )] for s > a.

Proof: By definition,

.C[u(x - c)J(x - c)] = 1= u(x - c)f(x - c)e-sx dx = 1= f(x - c)e-sx dx.

Making the change of variable ~ = x - c, we find

.C[u(x - c)f(x - c)] = 1= /(Oe-s(~+c) d~

= e-cs 1= f(~)e-s~ d~ = e-cs .C[f(x )] for s > a.

EXAMPLE 1 Using the Second Translation Property

to Calculate a Laplace Transform

Calculate the Laplace transform of

{

cosx,

f(x) =

0,