Explanation Given: The boundary problem is y ″ + y = x 2 − 2 π x ; y ( 0 ) = 0 , y ′ ( π ) = 0 Approach: For the boundary problem L [ y ] + μ r y = f ; B [ y ] = 0 Find the eigenfunctions { ϕ n } n = 1 ∞ and eigenvalues { λ n } n = 1 ∞ for the homogeneous equation corresponding to the given equation. Find the coefficients γ n for the eigenfunction expansion by computing f ( x ) r ( x ) = ∑ n = 1 ∞ γ n ϕ n If μ ≠ λ n for all n then the solution ϕ is given by ϕ = ∑ n = 1 ∞ γ n μ − λ n ϕ n If μ = λ n and γ n ≠ 0 for some n then, solution doesn’t exist. If μ = λ n and γ n = 0 for some n = N then ϕ = c N ϕ N + ∑ n = 1 n ≠ N ∞ γ n μ − λ n ϕ n Here, c N is arbitrary constant. Calculation: Consider the problem y ″ + y = x 2 − 2 π x The auxiliary equation is r 2 + 1 = 0 r 2 = − 1 The roots are r = ± i . So, the solution is y ( x ) = c 1 e i x + c 2 e − i x = c 1 ( cos x + i sin x ) + c 2 ( cos x − i sin x ) = ( c 1 + c 2 ) cos x + i ( c 1 − c 2 ) sin x = C 1 cos x + C 2 sin x It is given that y ( 0 ) = 0 , y ′ ( π ) = 0 . Therefore, C 1 cos ( 0 ) + C 2 sin ( 0 ) = 0 C 1 + 0 = 0 C 1 = 0 And C 1 cos ( π ) − C 1 sin ( π ) = 0 C 2 cos π = 0 C 2 ≠ 0 The eigenfunctions are given by, ϕ n ( x ) = C n sin ( n − 1 2 ) x For regular strum-Lioville boundary value problem, r ( x ) = 1 . So, ∫ a b ϕ n 2 ( x ) r ( x ) = 1 Substitute the corresponding values to get,

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 11.5, Problem 7E

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Chapter 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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The eigenfunction expansion for the solution to the given boundary problem.

Solution Summary: The author explains the formal eigenfunction expansion for the solution to the given boundary problem.

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Chapter 11 Solutions

To find:

The eigenfunction expansion for the solution to the given boundary problem.