The solution of the eigenvalue problem from the previous problem is
A) $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$
B) $\lambda = n \pi / L , X ( x ) = \cos ( n \pi x / L ) , n = 1,2,3 , \ldots$
C) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$
D) $\lambda = ( n \pi / L ) ^ { 2 } , X ( x ) = \cos ( n \pi x / L ) , n = 0,1,2 , \ldots$
E) $\lambda = ( n \pi / L ) , X ( x ) = \sin ( n \pi x / L ) , n = 0,1,2 , \ldots$

Question

Quizplus · Accepted Answer

The correct solution involves $\lambda = (n\pi/L)^2$ and $X(x) = \sin(n\pi x/L)$ with $n = 1,2,3,\ldots$. This is characteristic of solutions to certain differential equations, such as the Sturm-Liouville problems, where $\lambda$ represents the eigenvalues and $X(x)$ the eigenfunctions. The square in $\lambda$ indicates that it is related to the square of the wave number, which is typical in problems involving vibrations or waves, and the sine function suggests boundary conditions that are zero at both ends of the domain.

The Solution of the Eigenvalue Problem from the Previous Problem λ=nπ/L,X(x)=sin⁡(nπx/L),n=1,2,3,…\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldotsλ=nπ/L,X(x)=sin(nπx/L),n=1,2,3,…

The Solution of the Eigenvalue Problem from the Previous Problem $\lambda = n \pi / L , X ( x ) = \sin ( n \pi x / L ) , n = 1,2,3 , \ldots$