Explanation Given: The boundary-value problem is y ″ + ( λ + 1 ) y = 0 , y ′ ( 0 ) = 0 , y ′ ( 1 ) = 0 . Calculation: In the given boundary value problem when λ + 1 = 0 or λ = − 1 the equation becomes y ″ = 0 . The general solution of which is y = c 1 x + c 2 . Differentiate the above general solution with respect to x then, y ′ = c 1 (1) Substitute x = 0 and y ′ ( 0 ) = 0 in the equation (1), c 1 = 0 For the non-trivial solution y = c 2 . Therefore, λ = − 1 is an Eigen value and corresponding Eigen value is y = 1 . When λ < 0 , assume that λ + 1 = − α 2 , where without loss of generality the number α is positive. Then the auxiliary equation is m 2 − α 2 = 0 with roots − α and α . So the general solution is y = c 1 cosh α x + c 2 sinh α x . Differentiate the above general solution with respect to x as, y = c 1 sinh α x + c 2 cosh α x (2) Substitute x = 0 and y ′ ( 0 ) = 0 in the equation (2), c 2 = 0 Substitute x = 1 and y ′ ( 1 ) = 0 in the equation (2), c 1 = 0 Therefore, the solution is y ( x ) = 0

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 5.2, Problem 16E

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The eigenvalues and eigenfunctions for the given boundary-value problem.

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Chapter 5.2, Problem 15E

Chapter 5.2, Problem 17E

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Ch. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A force...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...

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