1550078481-Ordinary_Differential_Equations__Roberts_

The Laplace Transf arm Method 265

!

sinx,

y(x)= 2-2cos(x-l)+sinx,

1 - 2cos(x -1) + cos(x - 2) + sin x,

-oo < x::::: 1

The reader should verify that this solution simultaneously satisfies equations
(3a), (3b), and (3c).

EXERCISES 5.4

1. Express each of the following functions in terms of unit step functions.

r

0 ::::: x < 1

a. fi(x)=

1, 1::::: x

{"

2:S:x<4

b. h(x) =

0 , otherwise

{ 0,

0::::: x < 1

c. h(x) =

(x-1)^2 , 1::::: x

{ 0,

0 ::::: x < 1

d. f4(x) =

x^2 - 2x + 3, 1::::: x

{ 0,

0 ::::: x < 7l'

e. f5(x) =

sin3(x - 7r), 7l'::::: x

r

0 ::::: x < 1

f. f5(x) =

1, 1::::: x

r

0 ::::: x < 1

g. h(x) =

0, 1::::: x

2. Find the Laplace transform of the functions in exercise 1.