I am having some trouble solving this differential equation for the rate of infected computers in a botnet at time t
$$\frac{\mathrm{d}x }{\mathrm{d} t} = \frac{1}{c\nu (1-x) + \beta x(1-x) - \gamma x}$$
with $$x(0) = 1$$ i.e at time 0, there is one computer in the botnet.
I am trying to get this in terms of x(t) but the algebra is becoming quite complex for me, can anyone assist?
Thanks!
Notice, use variable separable method as follows $$\frac{dx}{dt}=\frac{1}{c\nu(1-x)+\beta x(1-x)-\gamma x}$$ $$(c\nu(1-x)+\beta x(1-x)-\gamma x)\ dx=dt$$ $$c\nu\int \ dx+(\beta-c\nu-\gamma)\int x\ dx-\beta\int x^2\ dx=\int \ dt$$ $$c\nu x+(\beta-c\nu-\gamma)\frac{x^2}{2}-\beta\frac{x^3}{3}=t+C$$ now, setting $x=1$ at $t=0$, we get $$c\nu \cdot 1+(\beta-c\nu-\gamma)\frac{1^2}{2}-\beta\frac{1^3}{3}=0+C\implies C=c\nu+(\beta-c\nu-\gamma)\frac{1}{2}-\beta\frac{1}{3}$$ I hope you can take it from here