m257_316_midterm1_sec101w2008

m257_316_midterm1_sec101w2008 - di f erent cases for the...

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Math 257/316, Midterm 1, Section 101 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 )=1+2 x on the interval 0 <x< 3 . (a) Sketch the even and odd periodic extensions of f with period 6 . [5 marks] (b) Expand f ( x ) in a half range Fourier cosine series. [15 marks] (c) Use the result in (b) to derive a series expansion for π 2 8 . [10 marks] Hint: It may be useful to know that 3 Z 0 x cos( n π x 3 ) dx = ½ 0 if n is even 18 n 2 π 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 homogeneous boundary conditions (you need not give all the details of the
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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 ) = 0 = u ( , t ) IC : u ( x, 0) = 2 cos x sin x [20 marks] 3. Consider the second order di f erential equation: Ly = 2 xy 00 + (1 + 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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