1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Trans! orm Method 227

We now prove the Laplace transform is a linear operator.

LINEARITY PROPERTY OF THE LAPLACE TRANSFORM

The Laplace transform is a linear operator. That is, if Ji ( x) and h ( x) are

functions which have Laplace transforms for s > s 1 ands> s 2 , respectively,

and if c 1 and c2 are constants, then

Proof: Let s > max(s 1 , s 2 ). Then, by definition,

.C[cif1(x) + c2h(x)] = 1= {cif1(x) + c2h(x)}e-sx dx

= C1 1= Ji (x )e-sx dx + C21= h (x )e-sx dx

= c1.C[fi(x) ] + c2.C[h(x)].

EXAMPLE 4 Using the Linearity Property to Calculate a

Laplace Transform

Calculate .C[4x^2 + 3].

SOLUTION

Using the linearity property of the Laplace transform, we find

2 1 8 3

.C[4x^2 + 3] = 4.C[x^2 ] + 3.C[l] = 43 + 3-= 3 + - for s > 0.

s s s s

EXAMPLE 5 Using an Identity and Linearity

Calculate .C[sinh bx].

SOLUTION
1 1

Since sinh bx = -ebx - -e-bx, we find using the linearity property of the

2 2
Laplace transform,

l [

.C[sinhbx = .C - e^1 bx - -^1 e-bx] =-.Ce^1 [ bx] - -J._,^1 r [ e-bx]

2 2 2 2

1 1 1 1 b

• ---- ---- -- for s > lbl.


• 2 s - b 2 s + b - s^2 - b2