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

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

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

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