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

Multiple Choice

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

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