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mid12010 - ∂u ∂t = ∂ 2 u ∂x 2< x< π t> BC...

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Math 257/316, Midterm 1, Section 101/102 8 October 2010 Instructions. The exam lasts 55 minutes. Calculators are not allowed. A formula sheet is attached. 1. Consider the ODE 2 x 2 y 00 + ( x + α ) y 0 - ( x + 1) y = 0 , in which α is a constant. (a) Classify the point x = 0 (as ordinary point, regular singular point, or irregular singular point) depending on the value of α . [5 marks] (b) For α = 0, find two independent solutions (for x > 0) in the form of series about x = 0 (you need only write the first three non-zero terms in each solution). [20 marks] 1
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2. Consider the following heat equation problem with zero boundary conditions:
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Unformatted text preview: ∂u ∂t = ∂ 2 u ∂x 2 , < x < π, t > BC : u (0 ,t ) = 0 = u ( π,t ) IC : u ( x, 0) = f ( x ) (a) Apply the method of separation of variables, and find the solution if f ( x ) = 3 sin(2 x )-sin(4 x ). [15 marks] (b) Find the Fourier series of the 2 π-periodic function f ( x ) with f ( x ) = 3 x on-π ≤ x ≤ π . [5 marks] ( Hint: R π-π x sin( nx ) dx = 2 π n (-1) n +1 n = 1 , 2 , 3 ,.... ) (c) Use your answer to (b) to find the solution of the above heat equation problem with f ( x ) = 3 x . [5 marks] 2...
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