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.
Well, when it's known what is constant and what it's not, the key to solving this equation is observation that it's first order inhomogeneous linear equation:
$$ \frac{d\omega}{dt} + c \left (\frac{be^{-bt}}{ae^{-bt}-C_1} \right) \cdot \omega = f(t) $$
General solution of this ODE is a linear combination of any solution of it and general solution of corresponding homogeneous linear ODE:
$$ \frac{d\omega}{dt} + c \left (\frac{be^{-bt}}{ae^{-bt}-C_1} \right) \cdot \omega = 0 $$
Rewriting your already known solution:
$$ \omega = \color{blue}{ C_2\cdot \left|C_1 - ae^{-bt}\right|^{c/a}} + \color{green}{\left|C_1 - ae^{-bt}\right|^{c/a} \cdot \int_{0}^{t}{\frac{f(\tau)d\tau}{\left|C_1 - ae^{-b\tau}\right|^{c/a}}}} $$ blue expression is a general solution to corresponding homogeneous ODE and green expression is a some solution of original ODE obtained via the method of constants variation.