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Let u be Find all t

Let u be Find all t

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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Let u be the solution of the wave equation utt = T ?Uxx, u(0, x) = cos x, Ut(0, x) = 0. Find all t such that u(t, x) = 0 for all x E R. (Suggestion: write the solution using d'Alembert's formula and use trigonometric identities).
Transcribed Image Text:Let u be the solution of the wave equation utt = T ?Uxx, u(0, x) = cos x, Ut(0, x) = 0. Find all t such that u(t, x) = 0 for all x E R. (Suggestion: write the solution using d'Alembert's formula and use trigonometric identities).
Expert Solution
Step 1

Given

utt=π2.uxx

u(0,x)=cosx

ut(0,x)=0

u(t,x)=0 for all x∈ℝ

 

Step 2

 The d' Alembert's Formula is in the form of,

u(t,x)=g(x+ct)+f(x-ct)2

On applying the given solutions 

u(t,x)=cos(x+πt)+cos(x-πt)2

u(t,x)=cos(x) cos(πt)

 

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