Deriving a Linear ODE Solution with an Integrating Factor
Summary
This note explains how to derive a closed-form solution to a first-order linear differential equation arising in a discussion of William Feller’s work on singular diffusion. It starts from the equation for the changing quantity and uses a relation between the independent variable and time to rewrite the differential equation in a form with a known coefficient.
The key step is multiplying by a power of the expression involving the exponential term, so the left side becomes the differential of a product. Integrating then yields the stated solution, with an arbitrary integration constant. The answer assumes the relevant expression is positive to omit absolute-value notation; the original formula retains absolute values to cover sign cases. This is a worked derivation rather than an application to trading, and it does not discuss numerical methods or the broader diffusion model.
Key ideas
- A first-order linear differential equation can be solved by finding an integrating factor.
- The time-dependent coefficient determines a power function that makes the left side a product differential.
- Integrating the resulting equation produces the general solution and an arbitrary constant.
- The derivation assumes positivity when it temporarily omits absolute values.
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# Regarding "Two Singular Diffusion Problems" by William Feller
# Regarding "Two Singular Diffusion Problems" by William Feller
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 by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/17654
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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