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solutions3-test3

# solutions3-test3 - MATH 3705A Test 3 Solutions[Marks...

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MATH 3705A 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 = 59, the Fourier sine series of f ( x ) = 2 , 0 x < 2 0 , 2 x 3 converges to [4] (a) -2 (b) -1 (c) 0 (d) 1 (e) None of these Solution: (a) 2. The solution of Laplace’s equation u xx + u yy = 0 within the rectangle [4] 0 < x < 2 , 0 < y < 3, which satisfies the boundary conditions u (0 , y ) = 0 , u (2 , y ) = 0 , u ( x, 0) = x 2 , u ( x, 3) = 0 is: (a) u ( x, t ) = X n =1 a n sinh nπx 3 sin nπy 3 (b) u ( x, t ) = X n =1 a n sinh (2 - x ) 3 sin nπy 3 (c) u ( x, t ) = X n =1 a n sinh (3 - y ) 2 sin nπx 2 (d) u ( x, t ) = X n =1 a n sinh nπx 2 sin nπy 3 (e) None of the above Solution: (c) 3. The solution of the wave equation u xx = 1 c 2 u tt , 0 < x < L , which satisfies the bound- [6] ary conditions u (0 , t ) = u ( L, t ) = 0, is given by u ( x, t ) = X n =1 sin nπx L a n cos nπct L + b n sin nπct L . Find the solution u ( x, t ) of u xx = 1 9 u tt , 0 < x < 2, which satisfies the boundary con- ditions u (0 , t ) = u (2 , t ) = 0 and the initial conditions u ( x, 0) = 0 and u t ( x, 0) = 3 sin( πx ) - sin(3 πx ). Write down the complete solution

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solutions3-test3 - MATH 3705A Test 3 Solutions[Marks...

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