34
The Big Theorem on the Frobenius
Method, With Applications
At this point, you may have a number of questions, including:
1.
What do we do when the basic method does not yield the necessary linearly independent
pair of solutions?
2.
Are there any shortcuts?
To properly answer these questions requires a good bit of analysis — some straightforward and
some, perhaps, not so straightforward. We will do that in the next chapter. Here, instead, we
will present a few theorems summarizing the results of that analysis, and we will see how those
results can, in turn, be applied to solve and otherwise gain useful information about solutions to
some notable differential equations.
By the way, in the following, it does not matter whether we are restricting ourselves to
differential equations with rational coefficients, or are considering the more general case in
which the coefficients involve fairly arbitrary functions analytic at
x
0
. The discussion holds for
either case.
34.1
The Big Theorems
The Theorems
The first theorem simply restates results discussed earlier in lemmas 33.1 and 33.2, and in section
33.5.
Theorem 34.1 (the indicial equation and corresponding exponents)
Let
x
0
be a point on the real line. Then
x
0
is a regular singular point for a given secondorder,
linear homogeneous differential equation if and only if that differential equation can be written
as
(
x
−
x
0
)
2
α(
x
)
y
′′
+
(
x
−
x
0
)β(
x
)
y
′
+
γ (
x
)
y
=
0
where
α
,
β
and
γ
are all analytic at
x
0
with
α(
x
0
)
negationslash=
0
. Moreover:
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736
Frobenius II
1.
The indicial equation arising in the method of Frobenius to solve this differential equation
is
α
0
r
(
r
−
1
)
+
β
0
r
+
γ
0
=
0
where
α
0
=
α(
x
0
)
,
β
0
=
β(
x
0
)
and
γ
0
=
γ (
x
0
)
.
2.
The indicial equation has exactly two solutions
r
1
and
r
2
(possibly identical). And, if
α(
x
0
)
,
β(
x
0
)
and
γ (
x
0
)
are all real valued, then
r
1
and
r
2
are either both real valued
or are complex conjugates of each other.
The next theoremis “the big theorem” of the Frobenius method. It describes generic formulas
for solutions about regular singular points, and gives the intervals over which these formulas are
valid.
Theorem 34.2 (general solutions about regular singular points)
Assume
x
0
is a regular singular point on the real line for some given secondorder homogeneous
linear differential equation with real coefficients. Let
R
be the corresponding Frobenius radius
of convergence, and let
r
1
and
r
2
be the two solutions to the corresponding indicial equation,
with
r
1
≥
r
2
. Then, on the intervals
(
x
0
,
x
0
+
R
)
and
(
x
0
−
R
,
x
0
)
, general solutions to the
differential equation are given by
y
(
x
)
=
c
1
y
1
(
x
)
+
c
2
y
2
(
x
)
where
c
1
and
c
2
are arbitrary constants, and
y
1
and
y
2
are solutions that can be written as
follows
1
:
1.
In general,
y
1
(
x
)
= 
x
−
x
0

r
1
∞
summationdisplay
k
=
0
a
k
(
x
−
x
0
)
k
with
a
0
=
1
.
(34.1)
2.
If
r
1
−
r
2
is
not
an integer, then
y
2
(
x
)
= 
x
−
x
0

r
2
∞
summationdisplay
k
=
0
b
k
(
x
−
x
0
)
k
with
b
0
=
1
.
(34.2)
3.
If
r
1
−
r
2
=
0
(i.e.,
r
1
=
r
2
), then
y
2
(
x
)
=
y
1
(
x
)
ln

x
−
x
0
 + 
x
−
x
0

1
+
r
1
∞
summationdisplay
k
=
0
b
k
(
x
−
x
0
)
k
.
(34.3)
4.
If
r
1
−
r
2
=
K
for some positive integer
K
, then
y
2
(
x
)
=
µ
y
1
(
x
)
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 Summer '09
 Differential Equations, Equations, lim, Complex differential equation, Frobenius method, Regular singular point

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