Explanation The given two dimensional wave equation in polar coordinates is u r r + 1 r u r + 1 r 2 u θ θ = a − 2 u t t which can be rewritten as u r r + 1 r u r + 1 r 2 u θ θ = u t t a 2 . (1) Let u ( r , θ , t ) = R ( r ) Θ ( θ ) T ( t ) . Differentiate the above equation with respect to r , θ , and t separately as follows. u r = R ′ Θ T u θ = R Θ ′ T u t = R Θ T ′ Again differentiate with respect to r , θ , and t separately as follows. u r r = R ″ Θ T u θ θ = R Θ ″ T u t t = R Θ T ″ Substitute the above obtained values in equation (1). R ″ Θ T + 1 r R ′ Θ T + 1 r 2 R Θ ″ T = R Θ T ″ a 2 Now divide the above equation by R Θ T on both sides. R ″ Θ T R Θ T + 1 r R ′ Θ T R Θ T + 1 r 2 R Θ ″ T R Θ T = 1 a 2 R Θ T ″ R Θ T R ″ R + 1 r R ′ R + 1 r 2 Θ ″ Θ = 1 a 2 T ″ T Let R ″ R + 1 r R ′ R + 1 r 2 Θ ″ Θ = 1 a 2 T ″ T be equation (2). Then by the separation of constant, 1 a 2 T ″ T = − λ 2 and Θ ″ Θ = − μ 2 . Substitute 1 a 2 T ″ T = − λ 2 and Θ ″ Θ = − μ 2 in R ″ R + 1 r R ′ R + 1 r 2 Θ ″ Θ = 1 a 2 T ″ T

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 10.7, Problem 21P

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The ordinary differential equations satisfied by R(r),Θ(θ), and T(t).

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Chapter 10.7, Problem 20P

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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 ordinary differential equations satisfied by R ( r ) , Θ ( θ ) , and T ( t ) .

The ordinary differential equations satisfied by R(r),Θ(θ), and T(t).

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