Section 7.2 - Differential Equations Math 308 Spring 2008...

Differential Equations Texas A&M University
Math 308, Spring 2008
c circlecopyrt F. Dos Reis
Section 7-2
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