Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation?
I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ 2\pi$ and $x(0)=x(2k\pi)$ for $\forall k\in\mathbb Z$.
I guess that may use the following lemma:
Lemma: Consider the differential equation $x' = f (t , x)$ where $f(t, x)$ is continuously differentiable in $t$ and $x$. Suppose that $f (t + T, x) = f (t , x)$ for all t . Suppose there are constants $p$, $q$ such that $f (t , p) > 0, f (t , q) < 0$ for all $t$ then there is a periodic solution $x(t )$ for this equation with $p < x(0) < q$.
Realy, I consider $p=2$ and $q=0$ but $f(t,q)=-1-\cos t\leq 0$ and this inequality is not strictly.