II Series Solutions of Ordinary Differential Equations
Contents
1
Taylor Series
1
2
Analytic functions
2
3
Singular point and ordinary point
2
4
CauchyEuler Equations
4
5
The Frobenius Method
6
6
Bessel’s Equation
7
We have fully investigated solving second order linear differential equations with constant
coeﬃcients:
Ay
′′
+
By
′
+
Cy
= 0
,
where A,B,C are constants. Now we will explore how to find solutions to second order linear
differential equations whose coeﬃcients are not necessarily constant:
P
(
x
)
y
′′
+
Q
(
x
)
y
′
+
R
(
x
)
y
= 0
.
1
Taylor Series
Definition 1
The Taylor series about
x
0
of a function
f
(
x
)
is the series
∞
∑
n
=0
f
(
n
)
(
x
0
)
n
!
(
x
−
x
0
)
n
.
There exists
R
≥
0
such that the series is convergent in

x
−
x
0

< R
and divergent in

x
−
x
0

> R
. The number
R
is called
Radius of Convergence
. We have
f
(
x
) =
∞
∑
n
=0
f
(
n
)
(
x
0
)
n
!
(
x
−
x
0
)
n
,

x
−
x
0

< R.
Theorem 1
(Theorem 3 in the book)
R
= lim
n
→∞
a
n
a
n
+1
,
a
n
=
f
(
n
)
(
x
0
)
n
!
.
Example.
f
(
x
) =
1
3
−
2
x
=
1
1
−
2(
x
−
1)
=
∑
∞
n
=0
2
n
(
x
−
1)
n
. R
= lim
n
→∞
2
n
2
n
+1
=
1
2
.
1
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2
Analytic functions
Definition 2
A function
f
(
x
)
is analytic at
x
0
if
f
has Taylor series about
x
0
which con
verges to
f
(
x
)
in an interval containing
x
0
.
Example.
f
(
x
) =
1
1
−
x
is analytic at
x
= 0;
f
(
x
) =
√
x
is not analytic at 0, since
f
′
(0)
does not exist;
f
(
x
) =
e
x
is analytic at any
x
.
Remark: If
f
and
g
are analytic at
x
0
, then
cf
,
f
±
g
,
fg
,
f/g
(if
g
(
x
0
)
̸
= 0) are analytic
at
x
0
.
Remark: If
f
is analytic at
x
0
, then its Taylor series about
x
0
is unique.
Example:
1
1
−
x
=
∞
∑
n
=0
x
n
,

x

<
1
.
arctan
x
=
∫
1
1 +
x
2
dx
=
∫
(
∞
∑
n
=0
(
−
1)
n
x
2
n
)
dx
=
x
−
x
3
3
+
x
5
5
−
x
7
7
+
...,
−
1
< x <
1
.
3
Singular point and ordinary point
Consider the equation
P
(
x
)
y
′′
+
Q
(
x
)
y
′
+
R
(
x
)
y
= 0
.
If we divide two sides by
P
(
x
), then the equation is changed to
y
′′
+
p
(
x
)
y
′
+
q
(
x
)
y
= 0
.
(1)
Definition 3
(Definition 4 in the book) If both
p
(
x
)
and
q
(
x
)
are analytic at a point
x
0
,
then
x
0
is called an ordinary point. Otherwise, it is called a singular point.
Example. If
p
(
x
) and
q
(
x
) are polynomials, then any point is an ordinary point.
Example. The following equation has singular points
x
= 1
,
2:
y
′′
+
x
+
e
x
x
−
1
y
′
+
x
+ 1
x
−
2
y
= 0
.
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 Spring '08
 lee
 Taylor Series, Gate, ORDINARY DIFFERENTIAL EQUATIONS, Regular singular point, singular point

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