I've been deriving Bass diffusion model and keep consistently finding a different result than Bass' original answer. To make things worse, every single link in the Google results page just copies the Bass original solution, while I am finding the different one. You don't have to know the model, just let me show you the math part, so you can check my solution.
The premise:
We have two formulations of the hazard rate --- as Bayesian conditional probability and as a linear function:
\begin{equation} \frac{f(t)}{1-F(t)} = p + q F(t) \end{equation}
where $f(t) = \frac{dF(t)}{dt}$. This gives the following differential equation:
\begin{equation} \frac{dF(t)}{dt} = (1 - F(t)) (p + qF(t)) \end{equation}
My solution
Using Chain Rule, we rewrite this as:
\begin{equation} \int \frac{dF(t)}{(1-F(t))(p+qF(t))} = \int dt = t \label{eq:diff} \end{equation}
Notice that:
\begin{equation} \frac{1}{(1-F(t))(p+qF(t))} = \left( \frac{1}{p+q} \right) \left( \frac{q}{p+qF(t)} + \frac{1}{1-F(t)} \right), \end{equation}
substituting this to the above equation, implies:
\begin{equation} \int \frac{q}{p+qF(t)} dF(t) - \int \frac{-1}{1-F(t)} dF(t) = (p + q) t \end{equation}
integrating yields:
\begin{equation} \log (p + qF(t)) - \log (1 - F(t)) = (p + q) t \end{equation}
using log properties:
\begin{equation} \log \left( \frac{p + qF(t)}{1 - F(t)} \right) = (p + q) t \end{equation}
or:
\begin{equation} \frac{p + qF(t)}{1 - F(t)} = e^{(p+q)t} \end{equation}
cross producting the fraction yields:
\begin{equation} p + qF(t) = e^{(p+q)t} - e^{(p+q)t} F(t) \end{equation}
finally:
\begin{equation} (q + e^{(p+q)t}) F(t) = (e^{(p+q)t} - p) \end{equation}
or:
\begin{equation} F(t) = \frac{e^{(p+q)t} - p}{e^{(p+q)t} + q} \end{equation}
The problem:
But Bass, somehow found the following:
cancelling out my answer does not yield Bass' answer. The unnecessary $q$ is ruining everything.
Can you please help me with this inconsistency?
Thank you.