Explanation Given: The given equations are, ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 , 1 < r , − π ≤ θ ≤ π u ( 1 , θ ) = f ( θ ) , − π ≤ θ ≤ π Calculation: Consider the given equation, ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 ... ( 1 ) Consider the relation, u ( r , θ ) = R ( r ) T ( θ ) ... ( 2 ) Differentiate Equation ( 2 ) with respect to r . ∂ u ∂ r = R ′ ( r ) T ( θ ) ... ( 3 ) Differentiate Equation ( 3 ) with respect to r . ∂ 2 u ∂ r 2 = R ″ ( r ) T ( θ ) Differentiate Equation ( 2 ) with respect to θ . ∂ u ∂ θ = R ( r ) T ′ ( θ ) ... ( 4 ) Differentiate Equation ( 4 ) with respect to r . ∂ 2 u ∂ θ 2 = R ( r ) T ″ ( θ ) Substitute R ( r ) T ″ ( θ ) for ∂ 2 u ∂ θ 2 , R ″ ( r ) T ( θ ) for ∂ 2 u ∂ r 2 and R ′ ( r ) T ( θ ) for in ∂ u ∂ r Equation ( 1 ) . R ″ ( r ) T ( θ ) + 1 r R ′ ( r ) T ( θ ) + 1 r 2 R ( r ) T ″ ( θ ) = 0 r 2 R ″ ( r ) + r R ′ ( r ) R ( r ) = − T ″ ( θ ) T ( θ ) ... ( 5 ) Equate the ratios of Equation ( 5 ) to λ . r 2 R ″ ( r ) + r R ′ ( r ) − λ R ( r ) = 0 ... ( 6 ) T ″ ( θ ) + λ T ( θ ) = 0 ... ( 7 ) When λ > 0 the general solution of Equation ( 7 ) is, T ( θ ) = A cos ( λ θ ) + B sin ( λ θ )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 10.7, Problem 13E

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A solution of the given Dirichlet problem for an exterior domain.

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Chapter 10 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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A solution of the given Dirichlet problem for an exterior domain.

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A solution of the given Dirichlet problem for an exterior domain.

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