At the beginning of the year 2000, the country Freedonia had a population p of 100 million people. The birth rate is 4% per annum and the death rate is 2% per annum, compounded daily. That is, the births and deaths occur every day at a uniform fraction of the current population and the next day the number of births and deaths changes because the population changed the previous day. For example, every day the number of people who die is the fraction 0.02/365 of the total population at the end of the previous day (neglect leap-year effects). Every day 275 immigrants enter Freedonia.

Question

(a) Write a difference equation for the population at the beginning of the nth day after January 1, 2000 with the immigration rate as the excitation of the system.

(b) By finding the zero-excitation and zero-state responses of the system determine the population of Freedonia at the beginning of the year 2050.

Answer 1

Solution:

The difference equation is