Math 334 Lecture #23
§
6.1: The Laplace Transform
An Idea.
Is there an invertible linear “operator”
L{
y
(
t
)
}
=
Y
(
s
) that tranforms an
IVP
±
y
00
(
t
) +
ay
0
(
t
) +
by
(
t
) = 0
,
y
(0) =
y
0
, y
0
(0) =
y
0
0
,
into an algebraic equation like
(
s
2
+
as
+
b
)
Y
(
s
)

(
as
+
b
)
y
0

ay
0
0
= 0?
[This asks if it is possible to transform the operation of diﬀerentiation with respect to
the variable
t
into the operation of multiplication by a another variable
s
.]
If so, this algebraic equation is easily solved for the transform of the solution of the IVP,
Y
(
s
) =
(
as
+
b
)
y
0
+
ay
0
0
s
2
+
as
+
b
,
and then applying the inverse of the operator to
Y
(
s
) gives the solution of the IVP:
y
(
t
) =
L

1
{
Y
(
s
)
}
.
An Operator.
The
Laplace transform
of a function
f
(
t
) with domain
t
≥
0 is the
function
F
(
s
) deﬁned by the improper integral
F
(
s
) =
L{
f
(
t
)
}
=
Z
∞
0
e

st
f
(
t
)
dt
= lim
A
→∞
Z
A
0
e

st
f
(
t
)
dt
whose domain consists of those values of
s
for which the improper integral converges.
Which functions have a Laplace transform? Under what conditions on
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Smith
 Calculus, Algebra, Derivative, lim, Continuous function, dt

Click to edit the document details