To determine The eigenvalues and eigenfunctions of the boundary value problem y ″ + λ y = 0 , y ′ ( 0 ) = 0 , y ( π ) = 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 = ± 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 ) = − μ c 1 sin μ x + μ c 2 cos μ x . y ′ ( x ) = − μ c 1 sin μ x + μ c 2 cos μ x y ′ ( 0 ) = − μ c 1 sin μ ( 0 ) + μ c 2 cos μ ( 0 ) 0 = − μ c 1 sin 0 + μ c 2 cos 0 0 = 0 + μ c 2 0 = c 2 Substitute the values c 2 = 0 , x = π and y ( π ) = 0 in the equation y ( x ) = c 1 cos μ x + c 2 sin μ x . y ( x ) = c 1 cos μ x + c 2 sin μ x y ( π ) = c 1 cos μ π + ( 0 ) sin μ π 0 = c 1 cos μ π + 0 c 1 cos μ π = 0 Since the value of c 2 = 0 , the constant c 1 ≠ 0 . Hence, the value of cos μ π = 0 . That is, the values of μ are μ 1 = 1 2 , μ 2 = 3 2 , μ 3 = 5 2 , ... . The general form of μ is given by μ n = ( 2 n − 1 ) 2 , n = 1 , 2 , 3 , ... . The corresponding eigenvalues are given by λ n = ( 2 n − 1 ) 2 4 , n = 1 , 2 , 3 , ... . The eigenfunctions corresponding to the above eigenvalues are given by y n = cos ( ( 2 n − 1 ) x 2 ) , n = 1 , 2 , 3 , ... . 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 = ± 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

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 10.1, Problem 15P

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

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Chapter 10.1, Problem 14P

Chapter 10.1, Problem 16P

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