Neumann problems

Definition

X'' (x) + \lambda X(x) = 0 \quad _{0<x<c},\quad X'(0)=0,\quad X'(c)=0

If eigenvalue $\lambda ={\lambda }_n={\frac{n^2{\pi }^2}{c^2}}_{(n=0,1,2,\dots )}$ then eigen function $X_n(x)=\cos {\frac{n\pi x}{c}}_{(n=0,1,2,\dots )}$ If $\lambda \ne {\lambda }_n{}_{(n=0,1,2,\dots )}$ then $X(x)=0$ is the only solution of problem