m257_316_midterm1_sec104w2008

m257_316_midterm1_sec104w2008 - di f erent cases for the...

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Math 257/316, Midterm 1, Section 104 20 October 2008 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 and odd periodic extension of f with period 2 π . [5 marks] (b) Use the half-range Fourier cosine series expansion to derive a series expansion for | x | on π < x< π .Nowuseth isser iestowr itedownaser iesexpans ionfor g ( x )= ½ x + π 2 if π <x< 0 x + π 2 if 0 x π [15 marks] (c) Use the series expansion for | x | to derive a series expansion for π 2 8 . [10 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
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Unformatted text preview: di f erent cases for the eigenvalue problem): u t = 2 u x 2 , &lt; x &lt; , t &gt; BC : u (0 , t ) x = 0 = u ( , t ) x IC : u ( x, 0) = 2 sin 2 ( x ) [20 marks] 3. Consider the second order di f erential equation: Ly = 2 x 2 y 00 + 3 xy (1 + x 2 ) 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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