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a) O Show that the wave equation Ytt = a²Yzz where a is a constant, can be reduced to the form Yuv = 0 by the change of variables u = x + at and v= x - at b) Use the results in part (a) and show that Y(x, t) can be written as Y(x, t) = (x + at) + (x-at), where and are arbitrary functions. Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Y₁(x,0) = 0, -∞ 0. Using the form of the solution obtained in part (b), show that and must satisfy Jo(x) + y(x) = f(x), o'(x) - '(x) = 0.

a) O Show that the wave equation Ytt = a²Yzz where a is a constant, can be reduced to the form Yuv = 0 by the change of variables u = x + at and v= x - at b) Use the results in part (a) and show that Y(x, t) can be written as Y(x, t) = (x + at) + (x-at), where and are arbitrary functions. Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Y₁(x,0) = 0, -∞ 0. Using the form of the solution obtained in part (b), show that and must satisfy Jo(x) + y(x) = f(x), o'(x) - '(x) = 0.

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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3) Can you answer just a,b,c
Show that the wave equation Ytt = a²Y where a is a constant, can be reduced to the form Yuv = 0 by the change of variables u = x + at and v= x - at b) Use the results in part (a) and show that Y(x, t) can be written as Y(x, t) = (x + at) + (x-at), where and are arbitrary functions. Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Yt(x, 0) = 0, -∞ < x < ∞, t> 0. Using the form of the solution obtained in part (b), show that and must satisfy Jo(x) + y(x) = f(x), [g'(x) — &'(x) = 0. Solve the equations of part (c) for and y, and show that d) Y(x, t) = [f(x − at) + f(x + at)].
Transcribed Image Text:Show that the wave equation Ytt = a²Y where a is a constant, can be reduced to the form Yuv = 0 by the change of variables u = x + at and v= x - at b) Use the results in part (a) and show that Y(x, t) can be written as Y(x, t) = (x + at) + (x-at), where and are arbitrary functions. Now, consider the wave equation in part (a) in an infinite one dimensional medium subject to initial conditions Y(x,0) = f(x), Yt(x, 0) = 0, -∞ < x < ∞, t> 0. Using the form of the solution obtained in part (b), show that and must satisfy Jo(x) + y(x) = f(x), [g'(x) — &'(x) = 0. Solve the equations of part (c) for and y, and show that d) Y(x, t) = [f(x − at) + f(x + at)].
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