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5.7: SERIES SOLUTIONS NEAR A REGULAR
SINGULAR POINTS (CONTINUED)
KIAM HEONG KWA
Convention 1.
We will refer to several equations from section 5.6
very often. When we refer to these equations, we will prepend
1
their
labels by the section number. For instance, (4) from section 5.6 will be
referred to as (5.6.4).
We continue to indicate the construction of a second solution of
(5.6.4) when the exponents
r
1
and
r
2
at the singularity for the regular
singular point
x
0
= 0 diﬀer by an integer, i.e., when
(1)
r
1

r
2
=
N
for a nonnegative integer
N
. For convenience, we shall assume that
x >
0. The case
x <
0 can be dealt with by making the substitution
ξ
=

x
. We also assume suﬃcient convergence for all power series
involved in the sequel.
Case
r
1

r
2
=
N
= 0
.
Let
(2)
y
(
x,r
) =
x
r
∞
X
n
=0
a
n
(
r
)
x
n
,
where
a
n
,
n
≥
1, are assumed to satisfy the recurrence relation (5.6.14)
with
a
0
6
= 0. Then it follows from (5.6.12) that
(3)
x
2
∂
2
y
∂x
2
+
x
[
xp
(
x
)]
∂y
∂x
+ [
x
2
q
(
x
)]
y
=
a
0
F
(
r
)
x
r
=
a
0
(
r

r
1
)
2
x
r
.
In particular, this shows that
y
1
(
x
) =
y
(
r
1
,x
) is a formal solution of
(5.6.4) because
F
(
r
1
) = (
r
1

r
1
)
2
= 0. This is what we have indicated
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 Winter '11
 Kwa
 Differential Equations, Equations

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