To determine To solve: The boundary value problem y ″ + y = 0 , y ( 0 ) = 0 , y ( L ) = 0 . Explanation The given differential equation is y ″ + y = 0 . The characteristic equation corresponding to y ″ + y = 0 is m 2 + 1 = 0 . Compute the roots of m 2 + 1 = 0 as shown below. m 2 + 1 = 0 m 2 = − 1 m = ± i The roots are m = + i and − i . Since the characteristic equation has complex conjugate roots, the general solution of y ″ + y = 0 is given by y = c 1 cos x + c 2 sin x . Now use the given initial conditions y ( 0 ) = 0 , y ( L ) = 0 and evaluate c 1 and c 2 . Here, the complementary function is y ( x ) = c 1 cos x + c 2 sin x . Substitute x = 0 and y ( 0 ) = 0 in the complementary function y ( x ) = c 1 cos x + c 2 sin x as follows. y ( x ) = c 1 cos x + c 2 sin x y ( 0 ) = c 1 cos 0 + c 2 sin 0 0 = c 1 ⋅ 1 + c 2 ⋅ 0 0 = c 1 Therefore, the value of c 1 is 0. Substitute c 1 = 0 in the complementary function y = c 1 cos x + c 2 sin x . y = c 1 cos x + c 2 sin x y = ( 0 ) cos x + c 2 sin x y = c 2 sin x Substitute x = L and y ( L ) = 0 in the complementary function y ( x ) = c 2 sin x as follows

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 10.1, Problem 3P

To determine

To solve: The boundary value problem y″+y=0, y(0)=0, y(L)=0.

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Chapter 10.1, Problem 2P

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Elementary Differential Equations and Boundary Value Problems, Enhanced

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