Let P(t) be the population of a city after t years. In this city the death rate is alpha times the population size, the birth rate is beta times the population size and the immigration out of city is with rate m.
(a) Write a di fferential equation for the population. What is the general solution?
(b) If the initial population is 1,000,000 people, the birth rate divided by the population
size is .02 bigger than the death rate divided by the population size, and 100,000
people leave the city per year. What will be the population be after 2 years?
(c) Under this model and the values given in part b), will the population expand or
decline?
(d) If the 100,000 people leave the city per year, then for what value of the di erence
between and does the population stay constant?
Im not sure how to approach this problem.
What do they mean by the general solution?
for the differential equation im thinking that P(t) is just a function of the initial population times alpha plus the initial population times beta plus m. does that sound right?
once i get that I think i should be able to figure out the rest.
Any help is appreciated.
Best Answer
We find an expression for $\frac{dP}{dt}$, the rate of change of the population at time $t$. There are three items that contribute: births, deaths, and people leaving. births at time $t$ are occurring at the rate $\beta P$. Deaths are occurring at the rate $\alpha P$. We conclude that $$\frac{dP}{dt}=\beta P -\alpha P-m=(\beta-\alpha)P-m.$$ We are asked to find the general solution of this equation. I don't know the style you use for such things.
There are several possible approaches. One is to note that our DE is separable, and can be rewritten as $$\frac{dP}{(\beta-\alpha)P-m}=dt.$$ Integrate. We get $$\frac{1}{\beta-\alpha}\ln\left|(\beta-\alpha)P-m\right|=t+C.$$ Multiply both sides by $\beta-\alpha$, and take the exponential. We get $$(\beta-\alpha)P-m=D\ast e^{(\beta-\alpha)t} $$ for some constant $D$. Now we can solve explicitly for $P$. The general solution is $$P(t)=\frac{m}{\beta-\alpha}+Ee^{(\beta-\alpha)t},$$ where $E$ is an arbitrary constant.
The above is the general solution of the DE that you were asked for.
I would slightly prefer to make instead the change of variable $y=(\beta-\alpha)P-m$. Then $\frac{dy}{dt}=(\beta-\alpha)\frac{dP}{dt}=(\beta-\alpha)y$.
The equation $\frac{dy}{dt}=(\beta-\alpha)y$ is the familiar differential equation of exponential growth (if $\beta\gt \alpha$). We can write down the general solution from memory, and then use that to get an expression for $P(t)$.
The above is what you explicitly asked about. Now take over. If you have difficulties, please leave a message.