Systems of Differential Equations (Part 1. Introduction)

To this point we’ve only looked as solving single differential equations.  However, many “real life” situations are governed by a system of differential equations.  Consider the population problems that we looked at back in the modeling section of the first order differential equations chapter.  In these problems we looked only at a population of one species, yet the problem also contained some information about predators of the species.  We assumed that any predation would be constant in these cases.  However, in most cases the level of predation would also be dependent upon the population of the predator.  So, to be more realistic we should also have a second differential equation that would give the population of the predators.  Also note that the population of the predator would be, in some way, dependent upon the population of the prey as well.  In other words, we would need to know something about one population to find the other population.  So to find the population of either the prey or the predator we would need to solve a system of at least two differential equations.

The next topic of discussion is then how to solve systems of differential equations.  However, before doing this we will first need to do a quick review of Linear Algebra.  Much of what we will be doing in this chapter will be dependent upon topics from linear algebra.  This review is not intended to completely teach you the subject of linear algebra, as that is a topic for a complete class.  The quick review is intended to get you familiar enough with some of the basic topics that you will be able to do the work required once we get around to solving systems of differential equations.