In this chapter we examine the general first order differential equation y'=g(x,y) and discover that the graphical techniques of Chapter 1 (where g(x,y) is independent of y) and Chapter 2 (where g(x,y) is independent of x) apply with little or no modification. This is important, because, unlike the situation in Chapters 1 and 2, in general we cannot find solutions of y'=g(y), and even when we are able to do so in special cases, such solutions are not always useful. Because of this, finding numerical solutions of y'=g(y) is important. Thus, we introduce a simple method of obtaining a numerical solution of y'=g(y)—namely, Euler's method—and show how it can be improved. Sometimes numerical solutions are misleading and give incorrect conclusions. This leads to a discussion of period doubling and chaos. As an aid to determining the accurate behavior of solutions, we also give a theorem that allows us to compare the solution of a differential equation we are unable to solve explicitly with one that we can solve. Such alternative methods are also useful when it is difficult or impossible to graph implicit solutions. In all cases our primary goal is to fully understand the solution's behavior even though we do not necessarily have an explicit solution in terms of familiar functions.
