a. Explanation It is known that, the Laplace’s equation in two dimensions is u x x + u y y = 0 . Let x = r cos θ , and y = r sin θ . Obtain the first derivative of u with respect to r as follows. u r = δ δ r ( u ) = δ u δ x ⋅ δ x δ r + δ u δ y ⋅ δ y δ r = u x cos θ + u y sin θ Obtain the first derivative of u with respect to θ as follows. u θ = δ δ θ ( u ) = δ u δ x ⋅ δ x δ θ + δ u δ y ⋅ δ y δ θ = u x ( − r sin θ ) + u y ( r cos θ ) Obtain the second derivative of u with respect to r as follows. u r r = δ δ r ( u r ) = δ δ r ( u x cos θ + u y sin θ ) = u x x δ x δ r cos θ + u x y δ y δ r cos θ + u y x δ x δ r sin θ + u y y δ r δ r sin θ = u x x cos 2 θ + u x y sin θ cos θ + u y x cos θ sin θ + u y y sin 2 θ = u x x cos 2 θ + 2 u x y sin θ cos θ + u y y sin 2 θ That implies, u r r = u x x cos 2 θ + 2 u x y sin θ cos θ + u y y sin 2 θ (1) Obtain the second derivative of u with respect to θ as follows. u θ θ = δ δ θ ( u θ ) = δ δ θ ( u x ( − r sin θ ) + u y ( r cos θ ) ) = { u x x δ x δ θ ( − r sin θ ) + u x y δ y δ θ ( − r sin θ ) − u x r cos θ + u y x δ x δ θ ( r cos θ ) + u y y δ y δ θ ( r cos θ ) − u y r sin θ } b. To determine To show: The R and Z satisfies the equation r R ″ + R ′ + λ 2 r R = 0 , and Z ″ − λ 2 Z = 0 .

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Question

Chapter 10.8, Problem 18P

To determine

To show: The axially symmetric solutions satisfy urr+1rur+uzz=0

To determine

To show: The R and Z satisfies the equation rR″+R′+λ2rR=0, and Z″−λ2Z=0.

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Chapter 10.8, Problem 17P

Chapter 10.8, Problem 19P

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Chapter 10 Solutions

Elementary Differential Equations and Boundary Value Problems, Enhanced

Ch. 10.1 - In each of Problems 1 through 13, either solve the...Ch. 10.1 - Prob. 2P Ch. 10.1 - In each of Problems 1 through 13, either solve the...Ch. 10.1 - Prob. 4P Ch. 10.1 - In each of Problems 1 through 13, either solve the...Ch. 10.1 - In each of Problems 1 through 13, either solve the...Ch. 10.1 - In each of Problems 1 through 13, either solve the...Ch. 10.1 - Prob. 8P Ch. 10.1 - Prob. 9P Ch. 10.1 - In each of Problems 1 through 13, either solve the...

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The axially symmetric solutions satisfy u r r + 1 r u r + u z z = 0

To show: The axially symmetric solutions satisfy urr+1rur+uzz=0

To show: The R and Z satisfies the equation rR″+R′+λ2rR=0, and Z″−λ2Z=0.

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Chapter 10 Solutions