The Solution of the Eigenvalue Problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$

Multiple Choice

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ is

A) $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
B) $\lambda = n \pi / 2 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \sin ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
D) $\lambda = ( 2 n - 1 ) \pi / 2 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$
E) $\lambda = ( 2 n - 1 ) ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$