1
ECE-205 : Dynamical Systems
Homework #6
Due : Friday October 22 at noon
1) In this problem we will derive some useful properties of Laplace transforms starting from the basic
relationship
0
( ) ( ) st
X s x t e dt
a) Let’s assume
()xt
is a causal signal (it is zero for
0t
). We can then write
( ) ( ) ( )x t x t u t
to emphasize the
fact that
()xt
is zero before time zero. If there is a delay in the signal and it starts at time
0
t
, then we can write
the signal as
0 0 0
) ( ) )( (x t tx t t u t t
to emphasize the fact that the signal is zero before time
0
t
.
Using the definition of the Laplace transform and a simple change of variable in the integral, show that
0
00
( ) ( )) ( st
x t t u t t X s e
b) Using the results from part a, determine the inverse Laplace transform of
3
( 2)( 4
() )
s
e
ss
Xs
Answer:
2( 3) 4( 3)
(1( 3)
2
)tt
e e ux tt
c) Starting from the definition of the Laplace transform, show that
()
() dX s
tx t ds
.
d) Using the result from part c, and the transform pair
1
( ) ( )() atutt X sa
e sx
, and some simple
calculus, show that
2 3
342
1
( ) , ( ) , ( )
( ) (
26
) ( )
at at at
te u t t e u t t e u t
s a s a s a
