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?

Multiply both sides by $u(t)$ to get
$$\frac12 \frac d{dt} |u|^2 = a u [u-\sin(u)]+ u b \le (|a| + |b|) (|u|+1)^2 \le 2(|a| + |b|)(|u|^2+1).$$
Divide both sides by $2(|u|^2+1)$ to get
$$ \frac14 \frac d{dt} \log(|u|^2+1) \le |a| + |b| .$$
Integrate from $t = 0$ to $t = T$ to get
$$ |u(T)|^2 + 1 \le (|u_0|^2+1) \exp\left(4 \int_0^T |a| + |b| \, dt\right) < \infty .$$
