21d_99f_fe

21d_99f_fe - Bender Math 21D Final Exam 11:302:30 Thursday...

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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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