Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0 (see Equation (1) of Section 1.3). Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0. (Use P for P(t).) dP dt =

If [tex]P(t)[/tex] is the country's population at the instant [tex]t[/tex], its variation over a small time interval [tex]\textrm{d}t[/tex] is given by:[tex]\textrm{d}P = \dfrac{\textrm{d}P}{\textrm{d}t}\textrm{ d}t,[/tex]where [tex]\dfrac{\textrm{d}P}{\textrm{d}t}[/tex] is the instantaneous rate of growth. In the absence of both immigration and emigration, this rate of change is proportional to the population itself:[tex]\dfrac{\textrm{d}P}{\textrm{d}t} = kP, \quad\textrm{with } k>0.[/tex]When we consider only immigration at a constant rate, we get:[tex]\dfrac{\textrm{d}P}{\textrm{d}t} = r, \quad\textrm{with } r>0.[/tex]So over a small time interval [tex]\textrm{d}t[/tex], we have two contributions:On the one hand, the population will increase by [tex]kP\textrm{ d}t[/tex] on its own;On the other hand, the population will increase by [tex]r\textrm{ d}t[/tex] due to immigration.All in all, we get:[tex]\textrm{d}P = kP\textrm{ d}t + r\textrm{ d}t  \iff \dfrac{\textrm{d}P}{\textrm{d}t} = kP + r,[/tex]which is the differential equation we were looking for.