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Pu subject to the initial conditions u(1,0) = f(x), and du (z, 0) = g(x). a. Using the transformation { = 1 – d and 7ŋ = 1 + ct, show that the cannonical form of the one-dimensional wave equation is given by Pu 0, ƏÇən and find the general solution of this cannonical form. b. Show that the solution to the wave equation is u(z, t) = ;(z + ct) + f(z – ct)) +; c. Using the specific initial conditions to the wave equation du (r, 0) = sin(x), u(x, 0) = 0, and show that the solution to the wave equation is given by 1 u(1, t) = sin(x) sin(ct) 2c

Pu subject to the initial conditions u(1,0) = f(x), and du (z, 0) = g(x). a. Using the transformation { = 1 – d and 7ŋ = 1 + ct, show that the cannonical form of the one-dimensional wave equation is given by Pu 0, ƏÇən and find the general solution of this cannonical form. b. Show that the solution to the wave equation is u(z, t) = ;(z + ct) + f(z – ct)) +; c. Using the specific initial conditions to the wave equation du (r, 0) = sin(x), u(x, 0) = 0, and show that the solution to the wave equation is given by 1 u(1, t) = sin(x) sin(ct) 2c

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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22:03 ( Back MAT_3547_2021_INTEGRATED_... Question 3 Consider the one-dimensional homogeneous wave equation: Pu subject to the initial conditions u(x,0) = f(x), du (r, 0) = g(x). and a. Using the transformation { = x – ct and 7 = x + ct, show that the cannonical form of the one-dimensional wave equation is given by Pu = 0, and find the general solution of this cannonical form. b. Show that the solution to the wave equation is u(z, t) = ;(z + + ct) + f(x – ct)) + 1 g(s).ds 2c, c. Using the specific initial conditions to the wave equation и(т,0) — 0, аnd du (x,0) = sin(x), show that the solution to the wave equation is given by 1 u(x, t) = sin(x) sin(ct) 2c 2
Transcribed Image Text:22:03 ( Back MAT_3547_2021_INTEGRATED_... Question 3 Consider the one-dimensional homogeneous wave equation: Pu subject to the initial conditions u(x,0) = f(x), du (r, 0) = g(x). and a. Using the transformation { = x – ct and 7 = x + ct, show that the cannonical form of the one-dimensional wave equation is given by Pu = 0, and find the general solution of this cannonical form. b. Show that the solution to the wave equation is u(z, t) = ;(z + + ct) + f(x – ct)) + 1 g(s).ds 2c, c. Using the specific initial conditions to the wave equation и(т,0) — 0, аnd du (x,0) = sin(x), show that the solution to the wave equation is given by 1 u(x, t) = sin(x) sin(ct) 2c 2
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