I have a differential equation:
$$\frac{di}{dt} = \beta\langle k \rangle i(1-i)-\mu i$$
I am ok with how I get there. I am doing some reading on my own and I need to understand how, given that the initial condition is $i_{0} = i(t=0)$ , I get the general solution to be:
$$ i = (1 - \frac{\mu}{\beta \langle k \rangle})\frac{C e^{(\beta \langle k \rangle - \mu)t}}{1+Ce^{(\beta \langle k \rangle - \mu)t}} $$
If I separate the differential equation I get: $$\mu i \ di -\frac{di}{i} - \frac{di}{(1-i)} = \beta \langle k \rangle \ dt $$
Then I integrate both sides: $$\frac{\mu i^{2}}{2} - ln i + ln(1-i) + C = \beta \langle k \rangle t $$
Is this correct so far? because I have been trusting the process but I'm not getting to the correct answer. I am particularly unsure about the part with $\mu$ because when I use the initial condition to get C, substituting t=0 and $i_{0}=i$, I do not get what the book has for C. What they have for C is $$\frac{i_{0}}{(1-i_{0}-\frac{\mu}{\beta \langle k \rangle})}$$
You have a mistake, in that the $\mu idi$ term should really be $\mu i dt$ (i assume it comes from the second term of the r.h.s. of the original equation). So instead, start with the original equation, and just divide both sides by the r.h.s. to get:
$$\frac{di}{\beta(k)i(1-i)-\mu i}=dt,$$
and integrate this (e.g. by partial fractions).