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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
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 Spring '11
 Smith
 Algebra

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