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To use the concept of logistic growth, it is needed to identify the maximum or the limit of population. In this problem, I can't identify what will be the maximum of the population should I use.

The rate of natural increase of the people in a certain city is proportional to the population. If the population increases from $30,000$ to $45,000$ in $10$ years, when will the population be $52,000$?

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    $\begingroup$ The statement The rate of natural increase of the people in a certain city is proportional to the population is consistent with exponential growth, not with logistic growth. The problem statement provides one equation on the parameters, so that there are too many parameters to identify if logistic growth is assumed. However, the problem can be solved if exponential growth is assumed. $\endgroup$ – EditPiAf Apr 15 '19 at 8:57
  • $\begingroup$ If your logistic form is something like $y(t) = \frac{M}{1+ Ae^{-kt}}$ then you have the three constants $M,A,k$ to find. But you only have two observations, so there will be multiple solutions $\endgroup$ – Henry Apr 15 '19 at 9:27
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You have $\frac {dN}{dt}=kN$ . Integrate using limits given in first case and find value of k. Then use this value to find time when population is 52000. For time you have to set appropriate reference.

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    $\begingroup$ Why $\frac {dN}{dt}=kN$ rather than $\frac {dN}{dt}=kN\left(1-\frac{N}{M}\right)$ ? $\endgroup$ – Henry Apr 15 '19 at 9:29

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