Nonlinear Systems & Linearization · Challenge problem

A pendulum driven by a constant torque obeys $x' = y,\; y' = -\sin x + \tfrac12$. In the interval $0 < x < \pi$ it has a center at $x = \tfrac{\pi}{6}$ and a saddle at $x = \tfrac{5\pi}{6}$ (both with $y = 0$). The energy $E = \dfrac{y^2}{2} - \cos x - \dfrac{x}{2}$ is conserved. Find the energy of the saddle minus the energy of the center.