m257_316_midterm1_sec104w2007

# m257_316_midterm1_sec104w2007 - x = 0 = u ( π , t ) IC : u...

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Math 257/316, Midterm 1, Section 104 17 October 2007 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Let f ( x )= x on the interval 0 <x< π . (a) Sketch the even periodic extension of f with period 2 π . [5 marks] (b) Expand f ( x ) in a Fourier cosine series. [15 marks] Hint: It may be useful to know that π Z 0 x cos( nx ) dx = ½ 0 if n is even 2 n 2 if n is odd 2. Apply the method of separation of variables to determine the solution to the one dimensional heat equation with the following mixed boundary conditions: u t = 2 u x 2 , 0 <x< π ,t> 0 BC : u (0 ,t )
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Unformatted text preview: x = 0 = u ( π , t ) IC : u ( x, 0) = 3 cos( 5 2 x ) [40 marks] 3. Consider the second order di f erential equation: Ly = 4 x 2 y 00 − ( x 2 + x ) y + y = 0 (1) (a) Classify the points x ≥ (do not include the point at in f nity) as ordinary points, regular singular points, or irregular singular points. [5 marks] (b) Use the appropriate series expansion about the point x = 0 to determine two independent solutions to (1). You only need to determine the f rst three non-zero terms in each case. [35 marks] 1...
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## This note was uploaded on 04/06/2012 for the course MATH 257 taught by Professor Peirce during the Fall '08 term at The University of British Columbia.

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