Did I do everything correct until the last line? I'm not sure how to proceed afterwards, please help. Solve wave equatiun in one dimensi on22u(Xit)%3Dfor -D0

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Description: Did I do everything correct until the last line? I'm not sure how to proceed afterwards, please help. Solve wave equatiun in one dimensi on22u(Xit)%3Dfor -D0

One dimensional wave equation is given by ∂u(x,t)∂t2=c2∂u(x,t)∂x2⋯⋯(1) for -∞<x<+∞u(x,0)=e-ax…

Answered: Solve wave equatiun in one dimensi on…

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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Did I do everything correct until the last line? I'm not sure how to proceed afterwards, please help.

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Expert Solution

Step 1

One dimensional wave equation is given by

$\frac{\partial u (x, t)}{\partial t^{2}} = c^{2} \frac{\partial u (x, t)}{\partial x^{2}} \dots \dots (1)$

for $- \infty < x < + \infty$ $u (x, 0) = e^{- a |x|}$

To find the most general solution of (1),

$u (x, t) = (c_{1} \cos (λ x) + c_{2} \sin (λ x)) (c_{3} \cos (c λ t) + c_{4} \sin (c λ t)) \dots \dots (2)$

Assuming the boundary condition

(I) $u (0, t) = 0$

$\begin{array}{rcl} 0 & = & (c_{1} \cos (0) + c_{2} \sin (0)) (c_{3} \cos (c λ t) + c_{4} \sin (c λ t)) \\ \Rightarrow & c_{1} = 0 \end{array}$

From (2), we get

$u (x, t) = (c_{2} \sin (λ x)) (c_{3} \cos (c λ t) + c_{4} \sin (c λ t)) \dots \dots (3)$

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