本题目来源于试卷: Differential Equations，类别为 IB数学
[问答题]
A video streaming servhjt ,jrwf7m i0y0y/ 3gice /snj*f+ rg,re company are monitoring their market share in a region in which+/rj*n,fgre s they have recently commenced operations.
The number of households, N , they predict will subscribe to the streaming service can be modelled by the logistic differential equation
$\frac{\mathrm{d} N}{\mathrm{~d} t}=\frac{3 k N(L-N)}{2 L}$
where t is time measured in years and k, L are positive constants.
The constant L represents the total number of households in the region who could possibly subscribe to the streaming service.
1. Show that $\frac{\mathrm{d}^{2} N}{\mathrm{~d} t^{2}}=\left(\frac{3 k}{2 L}\right)^{2}(N)(L-N)(L-2 N)$ .
2. Hence show that the number of households subscribing to the streaming service is predicted to increase at its maximum rate when $N=\frac{L}{2}$ .
3. Hence determine the maximum value of $\frac{\mathrm{d} N}{\mathrm{~d} t}$ in terms of k and L .
Let N_{0} be the number of households who have subscribed to the streaming service at the time the company start monitoring their market share.
4. By solving the logistic differential equation, show that its solution can be expressed in the form
k$ t=\left(\frac{2}{3}\right) \ln \left(\frac{N\left(L-N_{0}\right)}{N_{0}(L-N)}\right)$
After 12 years, the number of subscribed households is predicted to be 4$ N_{0} $. It is known that L=5$ N_{0} $.
5. Find the value of k for this model.
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