Math 334 Lecture #22
§
5.3: Power Series Solutions, Part II
Representation Principle.
When can a general solution of
y
+
p
(
x
)
y
+
q
(
x
)
y
= 0
be represented by power series?
If
p
(
x
) and
q
(
x
) are analytic at
x
0
, then the general solution of the ODE is
y
=
∞
n
=0
a
n
(
x

x
0
)
n
=
a
0
y
1
(
x
) +
a
1
y
2
(
x
)
,
where
y
1
and
y
2
are linearly independent power series solutions each analytic at
x
0
with
a radius of convergence at least as large as the smallest of the radii of convergence of the
power series expansions for
p
and
q
about
x
0
.
[Recall that a function is analytic about
x
0
if it has a power series expansion about
x
0
with
a positive radius of convergence that equals the function on the interval of convergence.]
The power series solutions
y
1
and
y
2
are solutions of the IVPS with initial conditions
y
1
(
x
0
) = 1
,
y
1
(
x
0
) = 0
,
and
y
2
(
x
0
) = 0
,
y
2
(
x
0
) = 1
.
The Wronskian of
y
1
and
y
2
at
x
=
x
0
is 1, so that
y
1
and
y
2
are indeed linearly
independent on the interval of convergence.
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 Spring '11
 Smith
 Power Series, Taylor Series, Finding Power Series Solutions

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