To determine The eigenvalues and eigenfunctions of the boundary value problem y ″ − λ y = 0 , y ( 0 ) = 0 , y ′ ( L ) = 0 . Explanation The given differential equation is y ″ − λ y = 0 . Case 1: λ > 0 . Take λ = μ 2 , μ > 0 and rewrite the equation y ″ − λ y = 0 . That is, the given equation becomes y ″ − μ 2 y = 0 , μ > 0 . The corresponding characteristic equation is r 2 − μ 2 = 0 . Solve for r in the equation r 2 − μ 2 = 0 . r 2 − μ 2 = 0 r 2 = μ 2 r = ± μ So, the general solution of the differential equation y ″ − μ 2 y = 0 is y = c 1 cosh μ x + c 2 sinh μ x . Compute the derivative of the equation y = c 1 cosh μ x + c 2 sinh μ x . y = c 1 cosh μ x + c 2 sinh μ x y ′ = c 1 ( μ sinh μ x ) + c 2 ( μ cosh μ x ) y ′ = μ c 1 sinh μ x + μ c 2 cosh μ x Substitute 0 for x and y ( 0 ) = 0 in the equation y ( x ) = c 1 cosh μ x + c 2 sinh μ x . y ( x ) = c 1 cosh μ x + c 2 sinh μ x y ( 0 ) = c 1 cosh μ ( 0 ) + c 2 sinh μ ( 0 ) 0 = c 1 cosh 0 + c 2 sinh 0 0 = c 1 ( 1 ) + c 2 ( 0 ) 0 = c 1 Substitute c 1 = 0 , x = L and y ′ ( L ) = 0 in the equation y ′ ( x ) = μ c 1 sinh μ x + μ c 2 cosh μ x . y ′ ( x ) = μ c 1 sinh μ x + μ c 2 cosh μ x y ′ ( L ) = − μ ( 0 ) sinh μ ( L ) + μ c 2 cosh μ ( L ) 0 = 0 + μ c 2 cosh μ L c 2 μ cosh μ L = 0 Since cosh μ L ≠ 0 for any value of μ > 0 , the constant c 2 = 0 . Therefore, y = 0 is the only solution if λ > 0 . Case 2: λ < 0 . Take λ = − μ 2 , μ > 0 and rewrite the equation y ″ − λ y = 0 . That is, the given equation becomes y ″ + μ 2 y = 0 , μ > 0 . The corresponding characteristic equation is r 2 + μ 2 = 0 . Solve for r in the equation r 2 + μ 2 = 0 . r 2 + μ 2 = 0 r 2 = − μ 2 r = ± i μ So, the general solution of the differential equation y ″ + μ 2 y = 0 is y = c 1 cos μ x + c 2 sin μ x . Compute the derivative of the equation y = c 1 cos μ x + c 2 sin μ x . y = c 1 cos μ x + c 2 sin μ x y ′ = c 1 ( − μ sin μ x ) + c 2 ( μ cos μ x ) y ′ = − μ c 1 sin μ x + μ c 2 cos μ x Substitute 0 for x and y ( 0 ) = 0 in the equation y ( x ) = c 1 cos μ x + c 2 sin μ x

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 10.1, Problem 19P

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The eigenvalues and eigenfunctions of the boundary value problem y″−λy=0, y(0)=0, y′(L)=0.

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Chapter 10.1, Problem 18P

Chapter 10.1, Problem 20P

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Elementary Differential Equations and Boundary Value Problems, Enhanced

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