   Chapter 14.3, Problem 78E

Chapter
Section
Textbook Problem

# Show that each of the following functions is a solution of the wave equation utt = a2uxx.(a) u = sin(kx) sin(akt)(b) u = t/(a2t2 − x2)(c) u = (x − at)6 + (x + at)6(d) u = sin(x − at) + ln(x + at)

(a)

To determine

To show: The function u=sin(kx)sin(akt) is a solution of the wave equation utt=a2uxx .

Explanation

Given:

The function is, u=sin(kx)sin(akt) .

Proof:

Take the partial derivative of the given function with respect to t and obtain ut .

ut=t(sin(kx)sin(akt))=sin(kx)t(sin(akt))=sin(kx)cos(akt)(ak)(1)=aksin(kx)cos(akt)

Thus, ut=aksin(kx)cos(akt) . (1)

Take the partial derivative the equation (1) with respect to t and obtain utt .

2ut2=t(aksin(kx)cos(akt))=aksin(kx)t(cos(akt))=aksin(kx)[sin(akt)(ak)]=a2k2sin(kx)sin(akt)

Thus, the partial derivative, 2ut2=a2k2sin(kx)sin(akt) . (2)

Take the partial derivative of the given function with respect to x and obtain ux

(b)

To determine

To show: The function u=t(a2t2x2) is a solution of the wave equation utt=a2uxx .

(c)

To determine

To show: The function u=(xat)6+(x+at)6 is a solution of the wave equation utt=a2uxx .

(d)

To determine

To show: The function u=sin(xat)+ln(x+at) is a solution of the wave equation utt=a2uxx .

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