After working through J.D Murray's Mathematical Biology, I have come across this differential equation during a derivation of the SIR model.
$\frac{dR}{dt}$ = a$[N - S_0 + (\frac{S_0}{p}-1)R - \frac{S_0R^{2}}{2p^{2}}]$
(all constants except R)
Currently struggling the work through this integral with hyperbolics. The writer says to factor the RHS quadratic in R, and then integrate to get $R(t)$ but my knowledge of hyperbolics is rusty at best. I'm struggling to get a function that I am actually able to integrate. Is it just a case of completing the square proceeding to integrate?
The solution given is:
$R(t) = \frac{p^{2}}{S_0}[(\frac{S_0}{p}-1) + \alpha tanh(\frac{\alpha at}{2} - \phi)]$
Where $\alpha = [(\frac{S_0}{p} - 1)^{2} + \frac{2S_0(N-S_0)}{p^{2}}]^{\frac{1}{2}}$
$\phi = \frac{tanh^{-1}(\frac{S_0}{p} - 1)}{\alpha}$
If you clean up the notation, you will find that you have
$$\frac{dR}{dt}=A+BR+DR^2$$
where $A,B,D$ are constants. Separation of variables gives
$$\int \frac{1}{A+BR+DR^2} dR = t+C.$$
I think you are interested in the case when $A+BR+DR^2$ has distinct real roots, i.e. $B^2-4AD>0$. Then $A+BR+DR^2=(R-R_1)(R-R_2)$, so you perform partial fraction decomposition and you get a sum of two (real) logarithms. You can then combine these and solve for $R$; the issue of rewriting it in terms of tanh should probably be the subject of a separate question.