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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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)].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Fcb7ef4a5-607e-4684-bfcd-3372f899ab6f%2Fzd236s_processed.jpeg&w=3840&q=75)
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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