Differential Equations
Texas A&M University
Math 308, Spring 2008
c
circlecopyrt
F. Dos Reis
Section 72
Definition:
Let
f
be a function on [0
,
∞
). The Laplace transform of
f
is the function
F
defined by the integral
F
(
s
) =
integraldisplay
∞
0
e

st
f
(
t
)d
t
=
lim
N
→∞
integraldisplay
N
0
e

st
f
(
t
)d
t
The domain of
F
is all the values of
s
for which the integral exists. The
Laplace transform of
f
is also written
L {
f
}
.
Exercise 1.
Find the Laplace transforms of the following functions and give their domains.
•
f
(
x
) = 1.
•
f
(
x
) = e
3
x
.
•
f
(
x
) = 3 + e

2
x
.
•
f
(
x
) = sin
x
Theorem:
Let
f
1
and
f
2
be 2 functions whose Laplace transforms exist
for
s > α
and let
c
1
and
c
2
be two real constants. Then for
s > α
,
L {
f
1
+
f
2
}
=
L {
f
1
}
+
L {
f
2
}
L {
c
1
f
1
+
c
2
f
2
}
=
c
1
L {
f
1
}
+
c
2
L {
f
2
}
Exercise 2.
Let
f
(
x
) =
braceleftbigg
3
if
x
∈
[0
,
10]
e
x

10
if
x
∈
(10
,
∞
)
. Find the Laplace transform of
f
.
Definition:
A function
f
is said to be piecewise continuous on a finite
interval [
a, b
] if
f
(
t
) is continous at every point in [
a, b
] except possibly
for a finite number of points at which
f
has a jump discontinuity.
A function
f
is said to be piecewise continuous on [0
,
∞
) if
f
is piecewise
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 Spring '08
 comech
 Differential Equations, Equations, Derivative, Laplace, Continuous function, F. Dos Reis

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