Example 1

A country has a population of ten million at time t = 0, and constant annual birth rate b = 0.020 and death rate d = 0.015 per person. Find the population at time f.

The population satisfies the differential equation

The initial condition is

y = 10⁷ at t = 0.

The general solution is

y = Ce^0.005t.

Since at t = 0, 10⁷ = Ce⁷ = C the actual solution is

y = 10⁷ e^0.005t.