Infection model differential equation solved by partial fractions, with long-term limit

A disease spreads through a colony of rabbits. Let \(N\) be the number of infected rabbits, in thousands, \(t\) days after the outbreak is first observed.

The spread is modelled by 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 \(t=0\), \(1500\) rabbits are infected.

1. Write \(\frac{1}{-3+4x-x^2}\) as partial fractions of the form \(\frac{A}{x-1}+\frac{B}{3-x}\), where \(A\) and \(B\) are constants.[3]
2. Hence, show that \(\ln\left(\frac{N-1}{3-N}\right)=2kt-\ln p\), stating the value of the positive constant \(p\).[8]
3. Express \(N\) in terms of \(k\) and \(t\), in the form \(N=\frac{a+be^{-2kt}}{1+ce^{-2kt}}\), where \(a\), \(b\) and \(c\) are integers.[5]
4. In the long term, the model predicts that the number of infected rabbits tends to a limiting value, \(L\).
   Find \(L\).[2]