Applications of Convolution
The Variation of Parameters Formula
In the theory of linear inhomogeneous differential equations
it is shown that the particular solution
y
(
x
) of the differential
equation
a
0
d
n
y
dx
n
+
a
1
d
n

1
y
dx
n

1
+
· · ·
+
a
n

1
dy
dx
+
a
n
y
=
f
(
x
)
with zero initial conditions, i.e.,
y
(0) = 0
, y
0
(0) = 0
,
· · ·
, y
(
n

1)
(0) = 0
,
is given by the socalled
Variation of Parameters
formula:
y
(
x
) =
Z
x
0
W
(
x, ξ
)
f
(
ξ
)
dξ,
wherein the kernel function,
W
(
x, ξ
), is given in terms of a cer
tain
Wronskian
determinant. This determinant is rather labori
ous to compute for even modest values of
n
, such as 4 or 5. Using
the Laplace transform we can obtain a very simple formula for
W
(
x, ξ
) which is computable with very modest effort.
Application of the Laplace transform to the differential equa
tion above gives
p
(
s
) (
L
y
) (
s
) = (
L
f
) (
s
);
p
(
s
) =
a
0
s
n
+
a
1
s
n

1
+
· · ·
+
a
n

1
s
+
a
n
.
Then
(
L
y
) (
s
) =
1
p
(
s
)
(
L
f
) (
s
)
.
1
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Suppose
p
(
s
) has distinct roots
λ
k
,
k
=
1
,
2
,
· · ·
, n
; real or
complex. Then we can write
1
p
(
s
)
=
c
1
s

λ
1
+
c
2
s

λ
2
+
· · ·
+
c
n
s

λ
n
=
n
X
k
=1
c
k
s

λ
k
,
where the
c
k
are given by the
residue formula
c
k
=
1
p
0
(
λ
k
)
.
Since the Laplace transform carries convolution products into
ordinary products of functions (of
s
), the inverse Laplace trans
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 Spring '10
 ROBINSON
 Equations, Laplace

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