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Bender
Math 21D
Final Exam
11:30–2:30 Thursday 12/16/99
•
Please put your name, ID number, and section number (or time) on your blue book.
•
The ﬁrst page of your blue book may contain notes. No other paper is allowed.
•
Calculators are NOT allowed.
•
You must show your work to receive credit.
1. (20 pts) Consider the series
∞
X
n
=0
(
x

2)
n
n
+1
.
(a) For what values of
x
is the series absolutely convergent? Your answer should be
an interval for
x
; that is, an expression like
a
≤
x<b
,or
a<x<b
, etc.
(b) For what values of
x
is the series conditionally convergent?
2. (60 pts) Solve the following diﬀerential equations. If initial conditions are given, ﬁnd
the particular solution. If there are no initial conditions, ﬁnd the general solution.
(a)
y
0
=
2
x

y
x
;
y
(0) = 2.
(b)
x
2
y
00
+3
xy
0

3
y
=0;
y
(1) = 4,
y
0
(1) = 0.
(c)
y
00

4
y
=16ln
t
.
You may leave integrals in your answer to (c).
3. (20 pts) Consider the diﬀerential equation
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 Fall '06
 Mohanty
 Math, Differential Equations, Equations

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