2
KIAM HEONG KWA
where
z
j
,
j
= 1
,
2
,
···
,n
, are the
n
zeros of
P
n
(
r
). Some of these zeros,
say
z
j
*
, may have multiplicity
>
1 in the sense that the factor
r

z
j
*
may appear more than once in (6). The
multiplicity
of
z
j
*
refers to the
number of times the factor
r

z
j
*
appears on the righthand side of
(6). Hence if
w
k
,
k
= 1
,
2
,
···
,m
, are the
distinct
zeros of
P
n
(
r
), then
(7)
P
n
(
r
) =
a
0
(
r

w
1
)
n
1
(
r

w
2
)
n
2
···
(
r

w
m
)
n
m
,
where
n
k
is the multiplicity of
w
k
for each
k
,
k
= 1
,
2
,
···
,m
.
Let
w
k
be a zero of
P
n
(
r
) with multiplicity
n
k
.
If
w
k
∈
R
, then
(8)
e
w
k
t
, te
w
k
t
,
···
, t
n
k

1
e
w
k
t
are solutions of
(1)
.
Clearly,
for any
c
l
∈
R
,
l
= 1
,
2
,
···
,n
k
,
(9)
n
k
X
l
=1
c
l
t
l

1
e
w
k
t
is a solution of
(1)
.
This is a consequence of the
principle of super
position
that says that any linear combination of solutions of a linear
homogeneous equation is again a solution of the same equation.
While
if
w
k
=
α
k
+
iβ
k
∈
C
, where
α
k
,β
k
∈
R
and
β
k
6
= 0
, then
(10)
e
α
k
t
cos(
β
k
t
)
, te
α
k
t
cos(
β
k
t
)
,
···
, t
n
k

1
e
α
k
t
cos(
β
k
t
)
,
e
α
k
t
sin(
β
k
t
)
, te
α
k
t
sin(
β
k
t
)
,
···
, t
n
k

1
e
α
k
t
sin(
β
k
t
)
,
are solutions of
(1)
, and so is
(11)
n
k
X
l
=1
t
l

1
e
α
k
t
[
A
l
cos(
β
k
t
) +
B
l
sin(
β
k
t
)]
for any real constants
A
l
and
B
l
,
l
= 1
,
2
,
···
,n
l
.
In fact, it can be
shown that the complex conjugate of
w
k
, i.e., ¯
w
k
=
α

iβ
, is also a zero
of
P
n
with the same multiplicity in this case. Furthermore, it is read
ily seen that ¯
w
k
gives rise to the same set of solutions to (1) as