1550078481-Ordinary_Differential_Equations__Roberts_

228 Ordinary Differential Equations

Another property of Laplace transforms which is useful in calculating the
transform of some functions is the following property call ed the translation
property.

TRANSLATION PROPERTY OF LAPLACE TRANSFORM

If .C[f(x)] = F(s) for s > s 0 , then .C[eax f(x)] = F(s - a) for s >so+ a.

Proof: By definition and hypothesis,

.C[f(x)] = 1

00

f(x)e-sx dx = F(s) for s >so.

Hence,

.C[eax f (x )] = loo eax f(x )e-sx dx

= 1

00

f(x)e-(s-a)x dx = F(s - a) for s - a> so.

That is,

.C[eax J(x)] = F(s - a) for s >so+ a.

As the following example illustrates, this property all ows us to calculate
easily the Laplace transform of the function eax f(x), if we already know the
transform of f(x).

EXAMPLE 6 Using the Translation Property

Calculate .C[xneax] where n is a positive integer.

SOLUTION

Earlier, we found that

n'

.C[xn] = sn~l = F(s) for s > 0.

Using the translation property, we now find that