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# A 10 points express f as a fourier series in the

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(a) (10 points) Express f as a Fourier series in the interval [ 1 , 1]. Leave the coeﬃcients indicated, do NOT evaluate any integrals. But anything that is 0 should not appear in the final expression. Fancy ways of writing 0 will be penalized. Solution. Because the function is even, only cosine terms will be present. Thus f ( x ) = 1 2 a 0 + n =1 a n cos nπx, where a n = 1 1 x 2 cos nπx dx. (b) (5 points) At what points x of the interval [ 1 , 1] is f ( x ) equal to the sum of its Fourier series? Solution. Because f is continuous at all points of the interval, and because f (1 ) = f (1+), the series converges to f ( x ) at ALL points of [ 1 , 1]. (c) (5 points) To what values, if any, does the Fourier series converge at the points x = 3 . 2 , 4 . 9, and 6? Solution. The function that is the sum of the Fourier series is periodic of period 2 and coincides with x 2 for | x | ≤ 1. Thus: Limit or sum at 3 . 2 = same as at 0 . 8, so to ( 0 , 8) 2 = 0 . 64. Limit or sum at 4 . 9 = same as at 0 . 9, so to 0 . 9 2 = 0 . 81. Limit or sum at 6 = same as at 0, so to 0 2 = 0. 2. Consider the following problem: (1 + x 2 ) y ′′ ( x ) + 2 xy + λ (1 + x 2 ) y = 0 , 0 < x < 1 , y (0) = y (1) = 0 .

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2 (a) (10 points) Write it out in the form of a regular Sturm–Liouville problem. What are p, q, σ ? Solution.
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