Consider Laplace's Equation on a Rectangle $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$

Consider Laplace's equation on a rectangle, $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$ with boundary conditions $u _ { x } ( 0 , y ) = 0 , u _ { x } ( 1 , y ) = 0 , u ( x , 0 ) = 0 , u ( x , 2 ) = f ( x )$ . When the variables are separated using $u ( x , y ) = X ( x ) Y ( y )$ , the resulting problems for $X$ and $Y$ are

A) $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0$
B) $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0$
C) $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , X ^ { \prime } ( 1 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 2 ) = 0$
D) $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$
E) $X ^ { \prime \prime } + \lambda X = 0 , X ^ { \prime } ( 0 ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0 , Y ( 2 ) = 0$