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The population model given by:

$$\frac{{dP}}{{dt}} = kP $$

(where k is a constant of proportionality.)

fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community it is assumed that the rate at which the population changes is a net rate—that is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t.

So I was able to come up with a suitable model, $$\frac{{dP}}{{dt}} = kP(t) - kP(t) $$

where

$$\begin{array}{l}b = kP(t)\\d = kP(t)\end{array} $$

for birth and death rate but the answer key numbers the proportionality constants as k1 and k2. Why couldn't you use the same k for your difference, especially if you use the same P(t)? It's a small thing but I could use some insight. Thanks in advance!

Books exact answer in solutions guide:

Let b be the rate of births and d the rate of deaths. Then $$\begin{array}{l}b = k_1P\\d = k_2P\end{array} $$

since

$$dP/dt = b - d $$ the differential equation is

$$dP/dt = k_1P - k_2P. $$

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You could take them equal, then the right side is zero. However, this is only one special case, there is nothing in the task description that links the birth and death rate.

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