Applications of Differential Equations
Populations tend to have a growth rate proportional to the present popluation. This is true whether we’re talking about a the population of a country or a colony of bacteria or a herd of deer.
If we call the population, P, we can write this as a differential equation:
dP/dt = kP
where k is the constant of proportionality. This constant k is a function of societial factors for the growth of a human population, but we can determine it if we have information about previous growth rates. Let’s work a problem.
Problem: The population of a certain country is 2 million and has doubled in the last 20 years. Find the expected population in 80 years.
First summarize the given information:
When t = 0 yrs P = 1 x 106
t = 20 yrs P = 2 x 106
t = 100 yrs P = ?
And dP/dt = kP
Now we separate the variables of our differential equation:
dP/P = k dt
Then integrate and solve for P:
ln P = kt + C
P = e kt + C
P = C e kt
Refer back to previous example if you are unclear on this step.
Now use the given information to solve for two constants, k and C.