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Second Order Equations] How do you solve 4? Question 4 is a continuation of the previous questions

Second Order Equations] How do you solve 4? Question 4 is a continuation of the previous questions

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Second Order Equations] How do you solve 4? Question 4 is a continuation of the previous questions

we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x <l utt - c²uxx = 0, with periodic boundary conditions and initial conditions u (-l, t) = u(l, t), ux (−l, t) = ux (l, t), (1) u (x,0) = (x), ut (x,0) = (x). 1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in class, show that there must exist a real number number A so that X" (x) + AX (x) = 0, T" (t) + c²AT (t) = 0. 2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ² for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or sin (¹7x). ηπχ 3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all possible separated solutions to obtain a guess for the general form of the solution u. 4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this general formula to the initial data & and 4. Using the formulas for Fourier series on -l < x < l, show that you can arrange the constants in your formula so that the function u determined by the formula obtains the initial data correctly.
Transcribed Image Text:we will use the method of separation of variables and Fourier series to derive a solution formula for the wave equation on -l < x <l utt - c²uxx = 0, with periodic boundary conditions and initial conditions u (-l, t) = u(l, t), ux (−l, t) = ux (l, t), (1) u (x,0) = (x), ut (x,0) = (x). 1. Suppose we have a "separated" solution u (x, t) = X(x)T(t). By following the method in class, show that there must exist a real number number A so that X" (x) + AX (x) = 0, T" (t) + c²AT (t) = 0. 2. Show that in order for X to satisfy the correct boundary conditions, we must have λ = (¹7) ² for n = 0,1,2,... and that we can take X to be any linear combination of cos (¹) or sin (¹7x). ηπχ 3. Solve the equation for T(t) (as we did in lecture) and then (just as in lecture) sum over all possible separated solutions to obtain a guess for the general form of the solution u. 4. By evaluating the formula (and the time derivative of the formula) at t = 0, relate this general formula to the initial data & and 4. Using the formulas for Fourier series on -l < x < l, show that you can arrange the constants in your formula so that the function u determined by the formula obtains the initial data correctly.
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