Analyze the periodic boundary value problem for the angular displacement of a circular ring along the same lines as the free case done in Example 11.3. Let 0 ≤ x ≤ 2n denote the angular coordinate along the ring, so that the boundary conditions are u (0) = u(2π) and (0) = u'(2π). Assume that the ring has constant stiffness c(x) = 1. Characterize the forcing functions that maintain equilibrium. Is this in accordance with the Fred holm alternative? Write down a forcing function that maintains equilibrium and determine the corresponding displacement and stress in the ring.