Chapter 7 3

7.
3 Operational Properties 1
First Translation Theorem (s-axis):
{ }
{ }
{ }

( ) ( )
( )
( )
at
s s a
s s a
e f t f t
F s
F s a

=
=
=
L L
Find the Laplace Transform of f (t) then replace s by s a
Formula no. 9 - 14
( ) F s a =
{ } { }
2 3 3
2
4
2
4
3

6

2
!
( )
t
s s
s s
e t t
s
s

=

=
`
)
=
L L
Example 1:
a = 2, f(t) = t
3
Replace s by s - 2
Can directly use formula no. 10 from table of Laplace
Transform:
{ }
1
!
( )
at n
n
n
e t
s a
+
=
L
a = 2, n = 3
{ }
{ }
5
5
2
5
2
3 3
3

9
3

5 9
sin sin
( )
t
s s
s s
e t t
s
s
+
+
=

=
`
+ )
=
+ +
L L
Example 2:
a = -5, f(t) = sin 3t
Replace s by s + 5
5 9 ( ) s + +
Transform:
{ }
2 2
sin
( )
at
k
e kt
s a k
=
+
L
a = -5, k = 3
{ } { }
{ }
2 2 2
2
2
2
1 1
2 1
2 2 1
=
( ) ( )
t
s s
s s
e t t
t t

=
= +

+
`
L L
L
Example 3:
a = 2, f(t) = (t - 1)
2
Replace s by s 2
3 2
2
3 2
2 2 1
=
2 2 1

2
2 2
( )
( ) ( )
s s
s
s s
s
s s

+
`
)
= +

Replace s by s 2
Inverse Laplace of First Translation Theorem:
{ } { }
1 1

( ) ( )
( )
s s a
at
F s a F s
e f t

=
=
L L
Find the inverse Laplace Transform of F(s) then multiply
together with e
at
(Formula no. 9 14)
1 1 3
4 4
1 1 1
3
1
!
( )
t t
e t e
s s

= =
` `
+ )

)
L L
Example 4:
a = -1
Transform:
{ }
1
1 1
1 1 !
!
( ) ( )
at n at n
n n
n
e t e t
n
s a s a
+ +

= =
`

)
L L
a = -1, n = 3
1
2
1
2
1
1 1
1
( )
( )
( )
s
s
s
s

`
+

)

+

=
`
+

)
L
L
Example 4:
Doesnt match any formula
term-wise division
1
2 2
1
2
1 1
1 1
1 1
1
1
( )
( ) ( )
( )
( )
s
s s
s
s
)

+

=
`
+ +

)

=
`
+
+

)
L
L
Now, can use
formula (6) and
formula (9)
(a = -1) (a = -1, n = 1)
1
2
1
2
6 11
3 9 11 ( )
s
s s
s
s

`
+ + )

=
`
+ +

)
L
L
Example 5:
Doesnt match any formula
cannot factor the
denominator, use
completing the square
1
2
1
2
3 2
3 3
3 2
( )
( )
( )
s
s
s
s

=
`
+ +

)

+

=
`
+ +

)
L
L
Look similar with formula
no. 12, but need to modify
1 1
2 2
3 3
3 2 3 2
( )
( ) ( )
s
s s

+

=
` `
+ + + +

) )
L L
Example 5 (continue):
Formula no. 12
a = -3 , k = 2
Formula no. 11
a = -3 , k = 2
3 1 3 1
2 2
3 3
1
3
2 2
3
2 2
2
cos sin
t t
t t
s
e e
s s
e t e t

=
` `
+ + ) )
=
L L
Second Translation Theorem (t-axis):
Definition 1: (Unit Step Function)
0 0
1
,
( )
,
t a
u t a
t a
<
The unit step function (or the Heaviside function) is defined as:
t
u(t a)
If a = 0, u(t) = 1
Piecewise Defined Function in terms of Unit Step
Function
A piecewise-defined function can be expressed in terms of unit
step function (single/compact form):
a t
a t
t h
t g
t f
<
=
0
) (
) (
) (
u(t 0) = 1
In unit step function:
or
f(t) = g(t)[u(t 0) u(t a)] + h(t)u(t a)
a t t h
) (
u(t 0) = 1
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
Example 1:
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
2
0
, 0 1
( )
, 1
t
f t
t
t
<

a = 1,
g(t) = 0
h(t) = t
2
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
Example 1:
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
2
0
, 0 1
( )
, 1
t
f t
t
t
<

a = 1,
g(t) = 0
h(t) = t
2
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
f(t) = 0[1 u(t 1)] + t
2
u(t 1)
= t
2
u(t 1)
Example 2:
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
sin , 0
( )
1 ,
t t
f t
t

<

a = ,
g(t) = sin t
h(t) = 1
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
Example 2:
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
sin , 0
( )
1 ,
t t
f t
t

<

a = ,
g(t) = sin t
h(t) = 1
f(t) = g(t)[1 u(t a)] + h(t)u(t a)
= sin t sin t u(t ) + u(t )
f(t) = sin t[1 u(t )] + u(t )
= sin t + [1 sin t]u(t )
Second Translation Theorem (t-axis):
Find the Laplace Transform of f (t) then multiply together
with e
-as
(Formula no. 20-21)
{ } { }
( ) ( ) ( ) ( )
as as
f t a u t a e f t e F s

= = L L
{ } { }
( ) ( ) ( )
as
g t u t a e g t a
= + L L
(Formula no. 22)
Use this formula if the independent variable in g(t) doesnt
match the independent variable in the unit step function.
{ }
1 ( )
s
e
u t
s
= L
Examples:
Formula no. 20
{ }
( )
as
e
u t a
s
= L
{ }
{ } { }
1
1
1
1
1
( )
( )
( ) ( )
s
t s t s
e
u t
s
e u t e e e
s

=
= =
L
L L
Examples:
Formula no. 20
Formula no. 21
{ }
( ) ( ) ( )
as
f t a u t a e F s
= L
{ }
( )
as
e
u t a
s
= L
Formula no. 21
{ }
( ) ( ) ( )
as
f t a u t a e F s
= L
{ }
{ } { }
1
1
1
1
1
( )
( )
( ) ( )
s
t s t s
e
u t
s
e u t e e e
s

=
= =
L
L L
Examples:
Formula no. 20
Formula no. 21
{ }
( ) ( ) ( )
as
f t a u t a e F s
= L
{ }
( )
as
e
u t a
s
= L
{ } { }
2
1 1
1 1 ( ) ( )
s s
t u t e t e
s
s

| |
= + = +
|
\
L L
Formula no. 21
Formula no. 22
{ } { }
( ) ( ) ( )
as
g t u t a e g t a
= + L L
{ }
( ) ( ) ( )
as
f t a u t a e F s
= L
Example: Find the Laplace Transform of the function
defined by:
2
0
, 0 1
( )
, 1
t
f t
t
t
<
From the previous example, we can write f(t) in unit

step function as:
a = 1,
g(t) = 0
h(t) = t
2
2
( ) ( 1) f t t u t =
defined by:
2
0
, 0 1
( )
, 1
t
f t
t
t
<

step function as:
a = 1,
g(t) = 0
h(t) = t
2
{ }
{ }
2
2
( ) ( 1)
( ) ( 1)
f t t u t
f t t u t
=
= L L
defined by:
2
0
, 0 1
( )
, 1
t
f t
t
t
<

step function as:
a = 1,
g(t) = 0
h(t) = t
2
{ }
{ }
{ }
{ }
2
2
2
2
( ) ( 1)
( ) ( 1)
( 1)
2 1
s
s
f t t u t
f t t u t
e t
e t t
=
=
= +
= + +
L L
L
L
Formula no. 22
defined by:
2
0
, 0 1
( )
, 1
t
f t
t
t
<

step function as:
a = 1,
g(t) = 0
h(t) = t
2
{ }
{ }
{ }
{ }
2
2
2
2
3 2
( ) ( 1)
( ) ( 1)
( 1)
2 2 1
2 1
s
s s
f t t u t
f t t u t
e t
e t t e
s
s s

=
=
= +
(
= + + = + +
(

L L
L
L
{ }
1
( ) ( ) ( )
as
e F s f t a u t a

= L
Inverse of Second Translation Theorem:
(Formula no. 21)
where:
{ }
1
( ) ( ) f t F s
= L
then find f(t a) and multiply together with unit step
function u(t a).
{ }
( ) ( ) f t F s = L
Example 1:
1 2
3
1
s
e
s

`
)
L
{ }
1
( ) ( ) ( )
as
e F s f t a u t a

= L
use formula no. 21
where:
3
1
2 ( ) , F s a
s
= =
By using this formula:
need to find f(t 2 );
1 2
3
1
2 2 ( ) ( )
s
e f t u t
s

=
`
)
L
Example 1:
1 2
3
1
s
e
s

`
)
L
{ }
1
( ) ( ) ( )
as
e F s f t a u t a

= L
use formula no. 21
where:
3
1 2 2
1
2
1 1 1
( ) , F s a
s
= =

= = =
Therefore:
1 2 2
3
1 1 1
2 2
2 2
( ) ( ) ( ) f t t f t t
s

= = =
`
)
L
1 2
3
2
1
2 2
1
2 2
2
( ) ( )
( ) ( )
s
e f t u t
s
t u t

=
`
)
=
L
Example 2:
1 2
2
4
s
s
e
s

`
+ )
L
{ }
1
( ) ( ) ( )
as
e F s f t a u t a

= L
use formula no. 21
where:
2
2
4
( ) ,
s
F s a
s
= =
+
Therefore:
1
2
2 2 2 2
4
( ) cos ( ) cos ( )
s
f t t f t t
s

= = =
`
+ )
L
1 2
2
2 2
4
2 2 2
( ) ( )
cos ( ) ( )
s
s
e f t u t
s
t u t

=
`
+ )
=
L
Example 3:
1
1 ( )
s
e
s s

`
+

)
L
use formula no. 21
1
1
1
1
( ) ,
( )
( ) ?
( ) ?
F s a
s s
f t
f t
= =
+
=
=
use partial fraction
Therefore:
1 ( ) ? f t =
1
1 1
1
1
( ) ( )
( )
[ ? ] ( )
s
e
f t u t
s s
u t

=
`
+

)
=
L
Example 3:
1
1 ( )
s
e
s s

`
+

)
L
use formula no. 21
1
1
1
1
1 1
1
1
( ) ,
( )
( )
t
F s a
s s
f t e
s s

= =
+

= =
`
+
)
L
Therefore:
1
1
1
1 1
( )
( )
( )
t
f t e
s s
f t e

= =
`
+
)
=
L
1
1
1 1
1
1 1
( )
( ) ( )
( )
[ ] ( )
s
t
e
f t u t
s s
e u t

=
`
+

)
=
L

Chapter 7 3

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7.

In unit step function:

In unit step function:

In unit step function:

In unit step function:

From the previous example, we can write f(t) in unit

From the previous example, we can write f(t) in unit

From the previous example, we can write f(t) in unit

From the previous example, we can write f(t) in unit

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