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Application of Laplace Transform
.

3 1
s = 1 : A = ; s = 5 : B = .
4 4
Similarly

C D
Y (s) = + ; C(s − 5) + D(s − 1) = 3,
s − 1 s − 5

3 3
s = 1 : C = − ; s = 5 : D = .
4 4
So,

 
3 1 1 1 3 1
t
 X(s) = +  x(t) = e + e 5t




 4 s − 1 4 s − 5  4 4
⇒
 3 1 3 1  3 3
 
  t 5t
 Y (s) = − +  y(t) = − e + e
4 s − 1 4 s − 5 4 4
. . . .
In this subsection we extend the definition of Laplace transform to
matrix-valued functions and apply this extension to solving systems of
differential equations. Let y (t), y (t), . . . , y (t) be members of PE. Consider the
2
n
1
vector-valued function
 
y (t)
1
 y (t) 
 2 
y(t) =  .  .
.
 . 
y (t)
n
The Laplace transform of y(t) is
∫   
∞ −st
y (t)e dt L[y (t)]
1
1
0 ∫
∫ ∞
∞  y (t)e −st   L[y (t)] 
dt

 .
L[y(t)] = y(t)e −st dt =  0 2 . .   2 . . 
 = 
0  .   . 
∫
∞ y (t)e −st dt L[y (t)]
0 n n
In a similar way, we define the Laplace transform of an m × n matrix to be the
m × n matrix consisting of the Laplace transforms of the component functions. If
the Laplace transform of each component exists then we say y(t) is Laplace
.
transformable.
Example 5.12, Find the Laplace transform of the vector-valued
function
 
2
t
1
y(t) =  
e t
. . . . .

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