Existence, Uniqueness & Bifurcations · Superchallenge problem

The logistic-type equation $y' = y - (1 + \cos t)\,y^2$ has exactly one solution $p(t)$ that is positive and $2\pi$-periodic. Find its time average $\dfrac{1}{2\pi}\displaystyle\int_0^{2\pi} p(t)\,dt$.