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) $X ^ { \prime \prime } + \lambda X = 0 , X ( 0 ) = 0 , X ( L ) = 0 , T ^ { \prime } + \lambda T = 0$

Question

Quizplus · Accepted Answer

Separating variables in a partial differential equation like $u _ { m } - u _ { t } = 0$ and applying the given boundary conditions leads to the equations for $X(x)$ and $T(t)$ as $X ^ { \prime \prime } + \lambda X = 0$ with $X(0) = 0, X(L) = 0$ for the spatial part, and $T ^ { \prime } + \lambda T = 0$ for the temporal part, without specific initial conditions for $T(t)$ given in the options.

Consider the Equation um−ut=0u _ { m } - u _ { t } = 0um​−ut​=0

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