Want to show that a solution of some ODE is bounded Suppose that $u(t)$ satisfies the differential equation 

$$\dot{u}(t)=a(t)[u(t)-\sin(u(t))]+b(t),\;u(0)=u_0$$

for all $t\in\mathbb R$. In addition suppose that $a,b$ are continuous integrable on $\mathbb R$. Now I want to show that $u(t)$ remains bounded on whole $\mathbb R$.
Since I am really not sure where to start, I wanted to ask if someone wants to give me some small hint?

Does not work for this general setting: if $a\equiv 0$ and $b(t)\ge1$, then $u$ is not bounded on $\mathbb R$.
Edit: Forgot about the integrability requirement.
You could try to use Gronwall's inequality.
http://en.wikipedia.org/wiki/Gronwall%27s_inequality
I suspect that you need $a(t)\le0$ to get the boundedness.
