In the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + x e ^ { t } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( L , t ) = 0 , u ( x , 0 ) = 0$ , the eigenvalues and eigenfunctions of the underlying homogeneous problem are
A) $\lambda = n ^ { 2 } , X = \sin ( n x )$
B) $\lambda = n ^ { 2 } , X = \cos ( n x )$
C) $\lambda = n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } , X = \sin ( n \pi x / L )$
D) $\lambda = n ^ { 2 } \pi ^ { 2 } / L ^ { 2 } , X = \cos ( n \pi x / L )$
E) $\lambda = n ^ { 2 } \pi ^ { 2 } , X = \sin ( n \pi x )$

Question

Quizplus · Accepted Answer

The correct eigenvalues and eigenfunctions are given by $\lambda = n ^ { 2 } \pi ^ { 2 } / L ^ { 2 }$ and $X = \sin ( n \pi x / L )$, which satisfy the boundary conditions $u(0,t) = 0$ and $u(L,t) = 0$. This is because the sine function becomes zero at both ends of the interval $[0, L]$ for integer values of $n$, matching the boundary conditions given.

In the Problem , the Eigenvalues and Eigenfunctions of the Underlying Homogeneous Problem