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