Fourier and Laplace
353
then
F s
s s
s
s
s
( )
( )
(
)
(
)
ϭ ϩ
ϩ
ϩ
5
2
3
10 3
3
5
2
2
7
2
2
2
2
+
+
+
+
+
b. Example (#2)
Let
F s
s
s
s
( ) ϭ
ϩ
ϩ ϩ
10 20
30
3
2
Find f(t).
ANALYTICAL Solution
(From Table 4.2)
f t
F s
f t
s
s
s
( ) £
( )
( )
ϭ
ϭ
ϭ
ϩ
ϩ ϩ
Ϫ
Ϫ
1
1
2
10 20
30
3
[
]
£ {
}
then
f(t) = 10 + 20t u(t) + 30e
−3t u(t)
c. Example (#3)
Let
F s
s
s
( ) ϭ
ϩ
ϩ
8
30
25
2
Find f(t).
ANALYTICAL Solution
(From Table 4.2)
F s
s
s
s
( ) ϭ ϩ
ϩ
ϩ
8
5
6
5
5
2
2
2
2

 

 
then
f(t) = £
−1 [F(s)] = 8 cos(5t) u(t) + 6 sin(5t) u(t)
d. Example (#4)
Let
Y s
s
s s
s
( )
(
)(
)
ϭ
ϩ
ϩ
ϩ
4
10
1
2
Find y(t).
ANALYTICAL Solution
Y s
A
s
B
s
C
s
( ) ϭ ϩ ϩ
ϩ ϩ
1
2
CRC_47760_CH004.indd 353
CRC_47760_CH004.indd 353
7/28/2008 12:25:59 PM
7/28/2008 12:25:59 PM