Consider the Equation $u _ { m } - u _ { t } = 0$

Multiple Choice

Consider the equation $u _ { m } - u _ { t } = 0$ with conditions $u ( 0 , t ) = 0 , u = ( L , t ) = 0 , u ( x , 0 ) = f ( x )$ . When separating variables with $u ( x , t ) = X ( x ) T ( t )$ , the resulting problems for $X , T$ are

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