   Chapter 14.3, Problem 75E

Chapter
Section
Textbook Problem

Verify that the function u = e − α 2 k 2 t sin k x is a solution of the heat conduction equation ut = α2uxx.

To determine

To verify: The function u=eα2k2tsin(kx) is a solution of the heat conduction equation ut=α2uxx .

Explanation

Verification:

The given function is, u=eα2k2tsin(kx) .

The equation ut=α2uxx is to be proved.

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

ux=x(eα2k2tsin(kx))=eα2k2tx(sin(kx))=eα2k2tcoskx(k)=keα2k2tcoskx

Thus, ux=keα2k2tcoskx . (1)

Take the partial derivative the equation (1) with respect to x and obtain uxx .

2ux2=x(keα2k2tcos(kx))=keα2k2tx(cos(kx))=keα2k2t[sin(kx)(

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