31
Power Series Solutions I: Basic
Computational Methods
When a solution to a differential equation is analytic at a point, then that solution can be rep
resented by a power series about that point. In this and the next chapter, we will discuss when
this can be expected, and how we might use this fact to obtain usable power series formulas for
the solutions to various differential equations. In this chapter, we will concentrate on two basic
methods — an “algebraic method” and a “Taylor series method” — for computing our power
series. Our main interest will be in the algebraic method. It is more commonly used and is the
method we will extend in chapter 33 to obtain “modified” power series solutions when we do
not quite have the desired analyticity. But the algebraic method is not well suited for solving
all types of differential equations, especially when the differential equations in question are not
linear. For that reason (and others) we will also introduce the Taylor series method near the end
of this chapter.
31.1
Basics
General Power Series Solutions
If it exists, a
power series solution
for a differential equation is just a power series formula
y
(
x
)
=
∞
summationdisplay
k
=
0
a
k
(
x
−
x
0
)
k
for a solution
y
to the given differential equation in some open interval containing
x
0
. The
series is a
general power series solution
if it describes all possible solutions in that interval.
As noted in the last chapter (corollary 30.10 on page 616), if
y
(
x
)
is given by the above
power series, then
a
0
=
y
(
x
0
)
and
a
1
=
y
′
(
x
0
)
.
Because a general solution to a firstorder differential equation normally has one arbitrary con
stant, we should expect a general power series solution to a firstorder differential equation to
also have a single arbitrary constant. And since that arbitrary constant can be determined by a
given initial value
y
(
x
0
)
, it makes sense to use
a
0
as that arbitrary constant.
8/14/2011
625
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626
Power Series Solutions I: Basic Computational Methods
On the other hand, a general solution to a secondorder differential equation usually has
two arbitrary constants, and they are normally determined by initial values
y
(
x
0
)
and
y
′
(
x
0
)
.
Consequently, we should expect the first two coefficients,
a
0
and
a
1
, to assume the roles of the
arbitrary constants in our general power series solutions for secondorder differential equations.
The Two Methods, Briefly
The basic ideas of both the “algebraic method” and the “Taylor series” are fairly simple.
The Algebraic Method
The
algebraic method
starts by assuming the solution
y
can be written as a power series
y
(
x
)
=
∞
summationdisplay
k
=
0
a
k
(
x
−
x
0
)
k
with the
a
k
’s being constants to be determined. This formula for
y
is then plugged into the
differential equation. By using a lot of algebra and only a little calculus, we then “simplify” the
resulting equation until it looks something like
∞
summationdisplay
n
=
0
bracketleftbig
n
th
formula of the
a
k
’s
bracketrightbig
x
n
=
0
.
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 Summer '09
 Differential Equations, Equations, Power Series, Taylor Series, Power Series Solutions

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