Regarding "Two Singular Diffusion Problems" by William Feller

Question

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.

Answer 1

For simplicity, we assume the necessary positivity, and then we can ignore the absolute signs. Note that \begin{align*} \big(C_1 - a e^{-bt} \big) d\omega = \big(C_1 - a e^{-bt} \big) f(t) dt + cbe^{-bt} \omega dt. \end{align*} That is, \begin{align*} \big(C_1 - a e^{-bt} \big)\,d\omega - cbe^{-bt} \omega \, dt= \big(C_1 - a e^{-bt} \big) f(t)dt. \end{align*} Then, \begin{align*} d\Big(\big(C_1-ae^{-bt} \big)^{-\frac{c}{a}} \, \omega\Big) &= -\frac{c}{a}\,\omega \big(C_1-ae^{-bt} \big)^{-\frac{c}{a}-1}\big(abe^{-bt}\big) dt + \big(C_1-ae^{-bt} \big)^{-\frac{c}{a}} d\omega\\ &=\big(C_1-ae^{-bt} \big)^{-\frac{c}{a}-1}\Big[\big(C_1-ae^{-bt} \big) d\omega- cbe^{-bt} \omega \, dt \Big]\\ &=\big(C_1-ae^{-bt} \big)^{-\frac{c}{a}}f(t)dt. \end{align*} The remaining is now obvious.