In this chapter we consider first order differential equations of the form dy/dx=g(y), where the right-hand side depends only on the dependent variable y. Such differential equations are called autonomous. They are important for two practical reasons: they are used to model many situations, and their properties form the basis for tackling more-complicated situations, which we will discuss in Chapters 6, 9, and 10.
One of the objectives of this chapter is to show you how the techniques developed in Chapter 1 can be applied to help us fully understand the qualitative and quantitative behavior of solutions of autonomous differential equations. These different techniques are needed because a single methodincluding analytical solutionscan lead to misleading conclusions, as we show in several examples.
In Chapter 1 we saw that only one solution of y'=g(x) could pass through a specified point (x0,y0) if g(x) is continuous. This guaranteed that solution curves did not intersect. We need to know what conditions must be imposed on g(y) so that the solution curves of y'=g(y) do not intersect. Thus, we need an existence and uniqueness theorem for initial value problems associated with y'=g(y). This is given, and its significance is very apparent when we hand-draw solution curves using information given by slope fields. Equilibrium solutions, phase line analysis, and bifurcation diagrams are new topics introduced in this chapter to help us determine properties of solution curves of y'=g(y).
We give examples of how autonomous differential equations are used in developing mathematical models for population dynamics, epidemics, and drug behavior. The exercises contain many more models of other situations. We use real data sets to evaluate the parameters that occur in the differential equations. This allows us to check the predictions of these mathematical models. It is vital to test a model's predictions against real data.