T3Solutions - MATH 3705C Test 3 Solutions[Marks Questions...

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MATH 3705C Test 3 Solutions March 16, 2007 Questions 1-2 are multiple choice. Circle the correct answer. Only the answer will be marked. [Marks] 1. At x = 58, the Fourier sine series of f ( x )= ½ 2 ; 0 x< 2 0 ; 2 x 3 ¾ converges, to [4] (a) 1 (b) -1 (c) 0 (d) -2 (e) None of these Solution: (b) 2. The solution of the wave equation u xx = 1 9 u tt ; 0 <x< 2, which satis¯es the boundary [4] conditions u (0 ;t )= u (2 ;t )=0 ,isg ivenby u ( x; t )= 1 X n =1 sin ³ n¼x 2 ´ a n cos μ 3 n¼t 2 + b n sin μ 3 n¼t 2 ¶¸ : If u ( x; t ) satis¯es the initial conditions u ( x; 0) = 0 and u t ( x; 0) = 3 sin( ¼x ) ¡ sin(3 ¼x ), the coe±cients a n and b n are given by (a) b 2 =3 ;b 6 = ¡ 1 ;b n = 0 otherwise, and a n = 0 for all n ¸ 1. (b) b 2 = 1 ¼ ;b 6 = ¡ 1 9 ¼ ;b n = 0 otherwise, and a n =0fora ll n ¸ 1. (c) a 2 = ¡ 3 ;a 6 =1 ;a n = 0 otherwise, and b n = 0 for all n
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T3Solutions - MATH 3705C Test 3 Solutions[Marks Questions...

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