When $u ( x , y ) = X ( x ) Y ( y )$

When $u ( x , y ) = X ( x ) Y ( y )$ is substituted into the equation $u _ { xx } + u _ { y y } = 0$ , the resulting equations for $X$ and $Y$ are

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