The Eigenvalue Problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0$

The eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi / 2 ) = 0$ has the solution

A) $y = \sin ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$
B) $y = \cos ( 2 n x ) , \lambda = 4 n ^ { 2 } , n = 1,2,3 , \ldots$
C) $y = \sin ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$
D) $y = \cos ( 2 n x ) , \lambda = 2 n , n = 1,2,3 , \ldots$
E) none of the above