Let us consider the following differential equation
$$
\dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}.
$$
> **My question.** Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the above equation be written in the form
$$
x(t) = at + r(a,t),
$$
where the term $r(a,t)$ is such that $r(a,t)\to 0$ for $a\to \infty$?


**PS:** Of course, the explicit solution can be computed via symbolic software tools, such as Wolfram Alpha. However, the symbolic expression (see [here][1]) looks quite messy and does not give much information on the behavior of the solution for $a\to\infty$...


  [1]: http://www.wolframalpha.com/input/?i=dx(t)%2Fdt%20%3D%20a%20-%20b*sin(x)