Have the following population model, $ \frac {dN}{dt}= cN(N-k)(1-N)$
The first stage of the question is to investigate the steady states, however im a little stuck on finding the solution to the differential equation given our initial condition $N(0)=2$ and assuming that $c=1$ and $k=0.5$,
$\int \frac {1}{N(N-0.5)(1-N)} dN = \int \left(\frac {-2}{N}+\frac{4}{N-0.5}+\frac{2}{1-N}\right) dN = \int dt$
$\displaylines{\Rightarrow -2\log(N)+4\log(N-0.5)-2\log(1-N)=t+{\rm constant} \Rightarrow \log \frac{2(N-0.5)}{N(1-n)}= t+c\cr \Rightarrow \frac{2(N-0.5)}{N(1-N)}=Ae^{t}\cr}$
Now there is a tip at the bottom of the question say consider a substitution of $k=N-0.5$.
$\Rightarrow$ $\frac{2(k)}{k+0.5(0.5-k)}$=$Ae^{t}$ $\Rightarrow$ $8k+4k^{2}Ae^{t}=Ae^{t}$
Therefore I must have gone wrong somewhere, any help would be much appreciated, many thanks.