# Systems and Signals Models - population model

1. Sep 7, 2010

### satchmo05

1. The problem statement, all variables and given/known data
At the beginning of the year 2000, country A had a population p of 100 million people. The birth rate is 4%/year and the death rate is 2%/year, compounded daily. Every day, 275 immigrants enter country A. Neglect leap-year effects.

Write a difference equation for the population at the beginning of the nth day after January 1,2000 (day 0) with the immigration rate as the input signal of the system.

By finding the zero-input and zero-state responses of the system, determine the population of country A at beginning of the year 2050.

2. Relevant equations
This is where I am super confused. I am looking at my text, and there isn't great help here. I have been going to class, nothing mentioned about it here. Is this simpler than I am making it out to be? I am thinking that in order to get the nth day of the population's output signal, I would need a summation of some sort. What is a difference equation?!?! That is my big question and how do I place this information into one to find the answer?

3. The attempt at a solution
I know this is not the final solution, but my summation model is:
4%/365 days = 0.01096% birth rate/day
-2%/365 days = -0.00548% death rate/day

($$\sum$$ from m=-infinity to n) [(1e8)(1+0.01096)m(1-0.00548)m + x[m]]

where x[m] = 275*m

However, solving the second part of this problem, I am finding that the population is around 2e50, which cannot be right! Please help! I appreciate all help in advance. Best.