Consider the following linear time-varying differential equation: $$\label{eq:1} \dot{x}(t)=-ax(t)+b\cos(\omega t)x(t), \quad x(0)=x_0\in\mathbb{R},\tag{$\star$} $$ where $a$, $b$, $\omega$ are positive scalars. It is well-known from basic control theory that, if we forget about the term $x(t)$ that post-multiplies the cosine term, namely, if we consider the simplified equation $$\label{eq:2} \dot{x}(t)=-ax(t)+b\cos(\omega t), \quad x(0)=x_0\in\mathbb{R},\tag{$\#$} $$ then the solution can be decoupled as a sum of a zero-input response and a forced response. More precisely, the forced response reads as $$ x_{\text{forced}}(t)=|G(j\omega)|\cos(\omega t +\angle G(j\omega)), $$ where $G(j\omega)=1/(1+ja\omega)$ is the frequency response of the system.
Now my question is: Can the solution of \eqref{eq:1} be somehow related to the frequency response of the system in \eqref{eq:2}? In particular, is it possible to rewrite \eqref{eq:1} as $$ \dot{x}(t)=-ax(t)+\beta(t), \quad x(0)=x_0\in\mathbb{R}, $$ where $\beta(t)\le b|G(j\omega)||x(t)|$?
I'm aware that my question is bit vague and some details are not formally stated. However, my principal aim is to gather some intuitions/ideas on a problem that I'm facing right now. So every comment/criticism/suggestion is more than welcome! Thanks a lot!