4038T3 - 1 2. Consider the equation u xx + u yy = 0 (with...

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Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books or notes are allowed, but a scientifc calculator is ok - though unnecessary . All work must be shown to receive credit. Please do not give decimal approximations For square roots, For trigonometric or exponential Functions, or For π . Maximum score = 100 points. 1. Consider y 0 + λy =0, y (0) = y (2 π ), and y 0 (0) = y 0 (2 π ). a. (5) Is this a Sturm-Liouville Problem? Show how you check the conditions. b. (20) ±ind all eigenvalues λ For the problem in (a) and all corresponding eigen- Functions. c. (5) Select two orthogonal eigenFunctions corresponding to the same eigenvalue, and evaluate the integral to show that these two eigenFunctions are orthogonal.
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Unformatted text preview: 1 2. Consider the equation u xx + u yy = 0 (with no boundary conditions). Use separation of variables to Fnd all solutions of the form u ( x, y ) = F ( x ) G ( y ). Be sure to consider each case for the constant k which arises in this method: a. (5) k = 0 b. (20) k = p 2 > 0. c. (5) k =-p 2 < 0. 2 3. (25) A vibrating string of length (with c 2 = 1) is described in general by u ( x, t ) = X n =1 ( A n cos nt + B n sin nt )sin nx Find u ( x, t ) if the initial displacement (at t = 0) is 2sin x-sin3 x and the initial velocity is sin3 x . 3 4. (15) Solve the hyperbolic equation u xx = u xy in the manner of DAlembert by substi-tuting v = y and z = x + y . 4...
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This note was uploaded on 01/06/2012 for the course MATH 4038 taught by Professor Staff during the Summer '08 term at LSU.

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4038T3 - 1 2. Consider the equation u xx + u yy = 0 (with...

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