Separable infection-model differential equation solved with partial fractions, and its limiting value

A model is used to describe how an infection spreads among the rabbits in a colony.

Let \(N\) be the number of infected rabbits, in thousands, \(t\) days after observations begin.

The model is the differential equation \(\frac{\mathrm{d}N}{\mathrm{d}t}=k(-3+4N-N^2)\), where \(k\) is a positive constant and \(t\geq0\).

When observations begin, \(1500\) rabbits are infected.

1. Write \(\frac1{-3+4x-x^2}\) in the form \(\frac{A}{x-1}+\frac{B}{3-x}\), stating the values of the real constants \(A\) and \(B\).[3]
2. Assume that \(1<N<3\). Separate the variables and use part (a) to show that \(\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p\), where \(p\) is a positive constant whose value should be found.[8]
3. Hence write \(N\) in terms of \(t\) and \(k\) in the form \(N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}\), where \(a\), \(b\) and \(c\) are integers whose values should be found.[5]
4. The model predicts that, as \(t\) increases without bound, the number of infected rabbits tends to a limiting value \(L\).
   Find \(L\).[2]