The Solution of the Eigenvalue Problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi ) = 0$

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi ) = 0$ is

A) $\lambda = n , y = \cos ( n x ) , n = 0,1,2 , \ldots$
B) $\lambda = n , y = \sin ( n x ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } , y = \cos ( n x ) , n = 0,1,2 , \ldots$
D) $\lambda = n ^ { 2 } , y = \sin ( n x ) , n = 1,2,3 , \ldots$
E) $\lambda = n , y = \cos ( n x ) + \sin ( n x ) , n = 1,2,3 , \ldots$

Correct Answer:

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