show that an ordinary differential equation has T periodic solution

Question

I have $dx/dt=-x^3(t)+h(t)$ where h(t) is a smooth, T-periodic function. Show that $x'(t)$ has a periodic solution.

So I tried solving the function as letting $h(t) =sin(t)$ and $cos(t)$ which are $2\pi$ periodic and was able to show that they have at least one periodic solution, but what should I do with a general function $h(t)$?

Answer 1

Hint: Let $x(t) = X(t,s)$ be the solution to your differential equation with initial condition $x(0) = s$. Then $X(T,s)$ is a continuous function of $s$. Show that for some $a < 0 < b$ we have $$a < X(T,a) < X(T,b) < b$$, and use the Intermediate Value Theorem to show there is some $s$ such that $X(T,s) = s$.