Each of these integrals is integrated by parts, then continuity of
f
(
t
)
collapses the end point evaluations and allows the single integral
noted on the right hand side, completing the general proof.
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Laplace Transforms: Part A
— (20/26)
Subscribe to view the full document.
Introduction
Laplace Transforms
Short Table of Laplace Transforms
Properties of Laplace Transform
Laplace Transform of Derivatives
Laplace Transform of Derivatives
3
Corollary (Laplace Transform of Derivatives)
Suppose that
1
The functions
f
,
f
0
,
f
00
, ...,
f
(
n

1)
are continuous and that
f
(
n
)
is piecewise continuous on any interval
0
≤
t
≤
A
2
The functions
f
,
f
0
, ...,
f
(
n
)
are of exponential order with

f
(
i
)
(
t
)
 ≤
Ke
at
for some constants
K
and
a
and
0
≤
i
≤
n
.
Then
L
[
f
(
n
)
(
t
)]
exists for
s > a
and satisfies
L
[
f
(
n
)
(
t
)] =
s
n
L
[
f
(
t
)]

s
n

1
f
(0)

...

sf
(
n

2)
(0)

f
(
n

1)
(0)
.
For our 2
nd
order differential equations we will commonly use
L
[
f
00
(
t
)] =
s
2
L
[
f
(
t
)]

sf
(0)

f
0
(0)
.
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Laplace Transforms: Part A
— (21/26)
Introduction
Laplace Transforms
Short Table of Laplace Transforms
Properties of Laplace Transform
Laplace Transform of Derivatives
Laplace Transform of Derivatives  Example
Example:
Consider
g
(
t
) =
e

2
t
sin(4
t
)
with
g
0
(
t
) =

2
e

2
t
sin(4
t
) + 4
e

2
t
cos(4
t
)
If
f
(
t
) = sin(4
t
), then
F
(
s
) =
4
s
2
+ 16
,
with
G
(
s
) =
4
(
s
+ 2)
2
+ 16
using the exponential theorem of Laplace transforms
Our derivative theorem gives
L
[
g
0
(
t
)] =
sG
(
s
)

g
(0) =
4
s
(
s
+ 2)
2
+ 16
However,
L
[
g
0
(
t
)]
=

2
L
[
e

2
t
sin(4
t
)] + 4
L
[
e

2
t
cos(4
t
)]
=

8
(
s
+ 2)
2
+ 16
+
4(
s
+ 2)
(
s
+ 2)
2
+ 16
=
4
s
(
s
+ 2)
2
+ 16
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Laplace Transforms: Part A
— (22/26)
Introduction
Laplace Transforms
Short Table of Laplace Transforms
Properties of Laplace Transform
Laplace Transform of Derivatives
Laplace Transform of Derivatives  Example
Example:
Consider the initial value problem:
y
00
+ 2
y
0
+ 5
y
=
e

t
,
y
(0) = 1
,
y
0
(0) =

3
Taking
Laplace Transforms
we have
L
[
y
00
] + 2
L
[
y
0
] + 5
L
[
y
] =
L
[
e

t
]
With
Y
(
s
) =
L
[
y
(
t
)], our derivative theorems give
s
2
Y
(
s
)

sy
(0)

y
0
(0) + 2 [
sY
(
s
)

y
(0)] + 5
Y
(
s
) =
1
s
+ 1
or
(
s
2
+ 2
s
+ 5)
Y
(
s
) =
1
s
+ 1
+
s

1
We can write
Y
(
s
) =
1
(
s
+ 1)(
s
2
+ 2
s
+ 5)
+
s

1
s
2
+ 2
s
+ 5
=
s
2
(
s
+ 1)(
s
2
+ 2
s
+ 5)
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Laplace Transforms: Part A
— (23/26)
Introduction
Laplace Transforms
Short Table of Laplace Transforms
Properties of Laplace Transform
Laplace Transform of Derivatives
Laplace Transform of Derivatives  Example
Example (cont):
From before,
Y
(
s
) =
s
2
(
s
+ 1)(
s
2
+ 2
s
+ 5)
An important result of the
Fundamental Theorem of Algebra
is
Partial Fractions Decomposition
We write
Y
(
s
) =
s
2
(
s
+ 1)(
s
2
+ 2
s
+ 5)
=
A
s
+ 1
+
Bs
+
C
s
2
+ 2
s
+ 5
Equivalently,
s
2
=
A
(
s
2
+ 2
s
+ 5) + (
Bs
+
C
)(
s
+ 1)
Let
s
=

1, then 1 = 4
A
or
A
=
1
4
Coefficient of
s
2
gives 1 =
A
+
B
, so
B
=
3
4
Coefficient of
s
0
gives 0 = 5
A
+
C
, so
C
=

5
4
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Laplace Transforms: Part A
— (24/26)
Introduction
Laplace Transforms
Short Table of Laplace Transforms
Properties of Laplace Transform
Laplace Transform of Derivatives
Laplace Transform of Derivatives  Example
Example (cont):
From the
Partial Fractions Decomposition
with
A
=
1
4
,
B
=
3
4
, and
C
=

5
4
,
Y
(
s
) =
1
4
1
s
+ 1
+
3
s

5
s
2
+ 2
s
+ 5
=
1
4
1
s
+ 1
+
3(
s
+ 1)

8
(
s
+ 1)
2
+ 4
Equivalently, we can write this
Y
(
s
) =
1
4
1
s
+ 1
+ 3
(
s
+ 1)
(
s
+ 1)
2
+ 4

4
2
(
s
+ 1)
2
+ 4
However,
L
[
e

t
] =
1
s
+1
,
L
[
e

t
cos(2
t
)] =
s
+1
(
s
+1)
2
+4
,
and
L
[
e

t
sin(2
t
)] =
2
(
s
+1)
2
+4
, so inverting the
Laplace transform
gives
y
(
t
) =
1
4
e

t
+
3
4
e

t
cos(2
t
)

e

t
sin(2
t
)
,
solving the initial value problem
Joseph M. Mahaffy,
h
[email protected]
 Fall '08
 staff