Fourier and Laplace
361
R.4.113 To gain additional experience, fi ve examples of DEs are presented and solved
manually, using the LT technique.
The following examples are solved for y(t) for each of the given DEs.
a. Example (#1)
Given the DE
2
4
0
0
2 0
dy t
dt
y t
y
( )
( )
( )
Ϫ
ϭ
ϭ
with IC (initial condition)
ANALYTICAL Solution
Taking the LT of the given equation results in
2
2 0
4
0
[ ( )
]
( )
sY s
Y s
Ϫ
Ϫ
ϭ
sY s
Y s
Y s
( )
( )
( )
Ϫ 20
4
2
2
ϭ
ϭ
then
sY s
Y s
( )
( )
Ϫ
Ϫ
2
2 0 0
ϭ
Y s s
( )(
)
Ϫ Ϫ
2
20 0
ϭ
then
Y s
s
( ) ϭ
20
2
Ϫ
taking the ILT
y t
e u t
t
( )
( )
ϭ 20
2
Ϫ
b. Example (#2)
Let the initial value of DE be
d y t
dt
y t
2
2
4
( )
( )
ϭ Ϫ
with the IC given by
y
dy t
dt
t
( )
( )
0
0
8
0
ϭ
ϭ
ϭ
and
ANALYTICAL Solution
Applying the LT to the preceding equation
s Y s
sy
dy t
dt
Y s
t
2
0
0
4
( )
( )
( )
( )
Ϫ
Ϫ
ϭ
ϭ
s Y s
s
Y s
2
0
8 4
( )
( )
( )
Ϫ
Ϫ ϭ
CRC_47760_CH004.indd 361
CRC_47760_CH004.indd 361
7/28/2008 12:26:02 PM
7/28/2008 12:26:02 PM