Fourier transform of the following time dependent expression

Question

I am working with the following expression which describes a simple Markov birth-death model of particle transport of uniform diameter:
\begin{gather*} \frac{\partial\eta(t)}{\partial t}=\nu+\mu\eta-\sigma\eta-w \quad (Eq. 1). \end{gather*} The variable $\eta$ is the number of moving particles for a given unit area $A$ at some time $t$, $\nu$ is the time varying inflow rate of particles to $A$, w is the time varying outflow rate of particles from $A$ and $\mu$ and $\sigma$ are rate constants of collective particle entrainment and particle deposition of particles that are found within the area $A$. I think the Fourier transform of Eq. (1) is:
\begin{gather*} i\omega\hat{\eta}(\omega)=\hat{\nu}+\mu\hat{\eta}-\sigma\hat{\eta}-\hat{w} \quad (Eq. 2), \end{gather*} where a circumflex $\hat{}$ denotes a Fourier transform. However, I am not sure that my treatment of the terms $\mu\eta$ and $\sigma\eta$ of Eq. (1) is correct. In searching for suggestions, I have reviewed the following question: Fourier Transform of Derivative.

Answer 1

For the Fourier Transform of a derivative you have a well known result: $$\mathcal {F} \left\{\frac {d^nf(t)}{dt^n}\right\}=(i\omega)^n\mathcal {F}\{f(t)\}(\omega)$$ For $\mu\eta$ and $\sigma\eta$ it's correct too.

You can write for simplicity: $$\mu\hat {\eta}-\sigma\hat {\eta}=(\mu−\sigma)\hat {\eta}$$