   Chapter 14.3, Problem 76E

Chapter
Section
Textbook Problem

Determine whether each of the following functions is a solution of Laplace’s equation uxx + uyy = 0.(a) u = x2 + y2(b) u = x2 − y2(c) u = x2 + 3xy2(d) u = ln x 2 + y 2 (e) u = sin x cosh y + cos x sinh y(f) u = e−x cos y − e−y cos x

(a)

To determine

Whether the function u=x2+y2 is a solution of Laplace’s equation, uxx+uyy=0 .

Explanation

The given function is, u=x2+y2 .

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

ux=x(x2+y2)=x(x2)+x(y2)=2x+0=2x

Thus, ux=2x . (1)

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

2ux2=x(2x)=2x(x)=2(1)=2

Thus, the partial derivative, 2ux2=2 .

Take the partial derivative of the given function with respect to y and obtain uy

(b)

To determine

Whether the function u=x2y2 is a solution of Laplace’s equation, uxx+uyy=0 .

(c)

To determine

Whether the function u=x3+3xy2 is a solution of Laplace’s equation, uxx+uyy=0 .

(d)

To determine

Whether the function u=lnx2+y2 is a solution of Laplace’s equation, uxx+uyy=0 .

(e)

To determine

Whether the function u=sinxcoshy+cosxsinhy is a solution of Laplace’s equation, uxx+uyy=0 .

(f)

To determine

Whether the function u=excosyeycosx is a solution of Laplace’s equation, uxx+uyy=0 .

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