Consider the Equation $u _ { x } + u _ { y y } = 0$

Multiple Choice

Consider the equation $u _ { x } + u _ { y y } = 0$ with conditions $u ( 0 , y ) = 0 , u = ( L , y ) = 0 , u ( x , 0 ) = f ( x ) , u ( x , H ) = 0$ . When separating variables with $u ( x , y ) = X ( x ) Y ( y )$ , the resulting problems for $X , Y$ are

A) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( H ) = 0$
B) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = 0$
C) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( 0 ) = 0$
D) $X ^ { \prime \prime } - \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } - \lambda Y = 0 , Y ( H ) = f ( 0 )$
E) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , Y ^ { \prime \prime } + \lambda Y = 0 , Y ( 0 ) = f ( x )$