The entire question reads:
Let $F$ satisfy the hypothesis of Lienard's Theorem. Show that $z'' + F(z') + z = 0$ has a unique, asymptotically stable, periodic solution. We're also given a hint to let $x = z'$ and let $y = -z$.
After some thought, I realize we probably want to get the equation to look like $x'' + F(x) x' + g(x) = 0$ (a system that we can apply Lienard's Theorem to), but I don't see how. Any help would be greatly appreciated!