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# HW7 - f for at least two periods(a f x = | x | − 1 ≤ x...

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Math 401 (Sec. 502) Spring, 2009 Homework 7 (due March 12) Q1. Use WKB method to obtain the large approximate eigenvalues of the following boundary value problems (where λ 1) (a) y ′′ + λ 2 e 4 t y = 0 , y (0) = y (1) = 0 (b) y ′′ + λ 2 (2 + t ) 2 y = 0 , y ( 1) = y (1) = 0 (c) y ′′ + λ 2 y = 0 , y (0) = y (1) = 0. PDE Q2. Find the Fourier series of each of the following functions. Sketch the graph of the periodic extension of
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Unformatted text preview: f for at least two periods: (a) f ( x ) = | x | , − 1 ≤ x ≤ 1 (b) f ( x ) = { − 1 if − 2 ≤ x ≤ 1 if 0 < x ≤ 2 (c) f ( x ) = x 2 , − π ≤ x ≤ π . Q3. Use the Fourier series of the function f in Q2.(c) to prove that (a) π 2 12 = 1 − 1 4 + 1 9 − 1 16 + ··· (b) π 2 6 = ∑ ∞ n =1 1 n 2 ....
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