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5.
If
( ) ( )
3
3 1
t
u t
e

=

and
( )
0
0
x
=
while
( )
0
0
x
=
d
, what is the total time solution for
( )
x t
?
(Although you could use
( ) ( )
3
3
3
3
3
0
3
5
8
t
t
G
G
e
e




=

to find the particular solution, we will
use Laplace Transform method to find the total solution.)
The Laplace Transform Table gives the transform pairs of
1
,
0
!
n
at
t e
t
n

≥
and
( )
1
1
n
s
a
+
+
.
If
0
n
=
and
0
a
=
the pair degenerates to a unit step and
1
s
.
If
0
n
=
and
3
a
=
the pair
degenerates to
3
t
e

and
1
3
s
+
.
Thus, by superposition, the Laplace Transform of
( ) ( )
3
3 1
t
u t
e

=

is
( )
( )
1
1
9
3
3
3
U s
s
s
s s
=

=
+
+
.
From Problem 1,
( )
( )
( )
0
0
2
2
2
2
5
2
5
U s
s
x
x
X s
s
s
s
s
+
+
=
+
+
+
+
+
d
.
As the initial conditions are given and the input has been rewritten in frequency space, the
desired output is
( )
( )
( )
( )
2
2
9
1
2
5
3
2
5
U s
X s
s
s
s s
s
s
=
=
+
+
+
+
+
Partial fraction expansion requires solving the
expression
( )
( )
2
2
9
1
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 Spring '08
 Perreira
 Laplace

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