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

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

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