The matrix eigenvalue problem leads a recurrence equation with constant coefficients $$ x_{n+1}-2 x_n +x_{n-1}= \Omega^2 x_n, \quad x_{N+1}= x_1 $$ As with all such constant-coefficient recurrence relations, you can try $x_n = e^{ikn}$ and so $\Omega^2$.

The matrix eigenvalue problem leads a recurrence equations with constant coefficients $$ x_{n+1}-2 x_n +x_{n-1}= \Omega^2 x_n, \quad x_{N+1}= x_1 $$ for the periodic case, and $$ x_{n+1}-2 x_n +x_{n-1}= \Omega^2 x_n, \quad x_{0}= x_{N+1}=0 $$ for the fixed end case.

As with all such constant-coefficient recurrence relations, you can try $x_n = e^{ikn}$ and so $\Omega^2$.