I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his research paper. More specifically, I don't understand how Feller solved $$dt=\frac{d\omega}{f(t)-cs\omega} \implies \frac{d\omega}{dt}=f(t)-cs\omega,$$ where $$\text{This is equation (3.4)}\,\,\,\,\,\,\,\,\,\,\,\,\,\, e^{-bt}\frac{as-b}{s}=C_1 \implies s=\frac{be^{-bt}}{ae^{-bt}-C_1},$$ and $a, b, C_1$ are constants with $b\neq0$ to get the solution$$\text{This is equation (3.5)}\,\,\,\,\,\,\,\,\,\,\,\, \omega = \left|C_1 - ae^{-bt}\right|^{c/a}\left\{C_2 + \int_{0}^{t}{\frac{f(\tau)d\tau}{\left|C_1 - ae^{-b\tau}\right|^{c/a}}}\right\},$$ where $C_2$ is a constant as well.
Please note that I have already verified this is true by differentiating it (and using the Fundamental Theorem of Calculus) but I don't understand how Feller derived it originally.
Can someone please explain to me in details or give me some hints regarding this? Any help will be greatly appreciated.