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SECTION 5.3 SOME THEORY FOR POWER SERIES SOLUTIONS
When
P
(
x
)
,Q
(
x
) and
R
(
x
) are polynomials – or even just when they can be represented
by power series – and
P
(
x
0
)
6
= 0 (so that a solution is guaranteed), we can plug
φ
(
x
) =
∞
X
n
=0
a
n
(
x

x
0
)
n
into the diﬀerential equation
P
(
x
)
y
00
+
Q
(
x
)
y
0
+
R
(
x
)
y
= 0
and determine the values of the
a
n
using a recurrence relation. Then to approximate values
of
y
(
x
) we can just add up a bunch of terms of the series.
But how do we know when to stop adding, and when we do, how do we know how close
we are to the exact value of
y
(
x
)? For example, we could stop adding when the next term
in the series is, say, less than half of some acceptable error.
EXAMPLE.
Suppose we do all this and we ﬁnd that
φ
(
x
) =
∞
X
n
=0
n
n
2
+ 1
x
n
. Our problem
needs the value
φ
(1) with an error of no more than 0.01. How many terms must we add up?
WRONG!!
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