1. Palm trees are being planted on an island. Let $N$ be the total number of palm trees planted on the island after $t$ days.

The variables $N$ and $t$ are modelled by the differential equation

$\frac{dN}{dt}=kN(N-1),N>1$

where $N>1$ and $k$ is a positive constant.

By solving the differential equation, show that

$N=\frac{1}{1-A{e}^{kt}}$

where $A$ is a positive constant.