Logistic Model

Consider the continuum logistic model that is described by the following differential equation:

$\frac{dN}{dt}=rN[1-\frac{N}{K}]$

$($a$)$ Describe the assumptions of the model and its constants. Show that the solution of the model is as follows:

$N(t)=\frac{K}{1+(\frac{K-N_0}{N_0})e^{-rt}},$

where $N_0$ is the initial condition.