Lecture 24
To begin, it may be helpful to make a few comments on the case of
regular
singular points
before starting the next big topic of the course, namely the
Laplace Transform.
If
x
=
x
0
is a regular singular point of a linear, second order ODE
(
∗
)
y
′′
+
p
(
x
)
y
′
+
q
(
x
) = 0
,
then we look for series solutions just like in the case of an ordinary point,
but with a diﬀerence, namely we try
y
=
x
r
∞
∑
n
=0
a
n
(
x
−
x
0
)
n
,
where
r
is a constant to be determined, which
r
need not be an integer,
or even a rational number. (When
r
is not rational, it will be a quadratic
irrational.) Assuming that the radius of convergence
R
(around
x
0
) is pos
itive, we diﬀerentiate the series expression term by term (for
x
satisfying

x
−
x
0

< R
) to get series expressions for
y
′
and
y
′′
. Plugging this informa
tion back in the diﬀerential equation, we get an identity of the form
x
r
∞
∑
n
=0
c
n
(
x
−
x
0
)
n
= 0
,
where the coeﬃcients
c
n
of the power series on the left are determined by the
coeﬃcients
a
n
of
y
(and hence of
y
′
, y
′′
) as well as by
r, p
(
x
)
.q
(
x
). It follows
that for such an identity to hold for all
x
close to
x
0
, we need to have
c
n
= 0
,
∀
n
≥
0
.
A key diﬀerence with the ordinary case occurs when we look at the condition
c
0
= 0, because this leads to an equation for
r
, called the
indicial equation
associated to (
∗
), which will be a quadratic equation in
r
. In this course
we will only look at the case when this has two distinct real roots, and
moreover assume that the two roots do not diﬀer by a integer. (As usual, it
is more subtle when there is a repeated root, and even when the roots are
not repeated, if they diﬀer by a positive integer, we will not be able to ﬁnd
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