5. Consider the wave equation Utt - Uxx=0 for 05 X 5 4 and t> 0 with u(0,x) = x/2 and ut(0, x) = -x/2 for 0 < x < 2 whil

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5 Consider The Wave Equation Utt Uxx 0 For 05 X 5 4 And T 0 With U 0 X X 2 And Ut 0 X X 2 For 0 X 2 Whil 1 (49.73 KiB) Viewed 60 times

5. Consider the wave equation Utt - Uxx=0 for 05 X 5 4 and t> 0 with u(0,x) = x/2 and ut(0, x) = -x/2 for 0 < x < 2 while u(0,x) = 2 - x/2 and u:(0, x) =- 2 + x/2 for 2 < x < 4. At x = 0, ux (t,0) = 0 and, at x = 4, uſt, 4) = 0. a) What are the characteristics of the equation? b) Solve for u(t,x) for 0 < x < 40< t < 8 using the D'Alambert solution method. c) Can you see the pattern and suggest the solution for Osts o? d) Solve using separation of variables and eigenfunction expansion. e) Solve using a Laplace transform. f) Compare the three solutions.