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Qualitative approach to autonomous equations.

In our courses, the analytic solution of differential equations definitely takes a back seat to qualitative and numerical techniques. At the outset, when covering first order equations, we remind students how to solve separable equations. But that is essentially the only analytic technique we include until the end of the first order discussion when we finally solve first order linear equations and mention changes of variables.

Most of our effort deals with autonomous equations of the form

dy / dt = f(y)

Although this type of equation is separable, we rarely take the time to carry out the integrations explicitly (when we do it is most often to show students how ``ugly'' and uninformative the resulting formulas are). Rather, we emphasize the interrelationships between four different pictures associated with this equation. These pictures are fairly easy to draw using only the formula for f(y), and which they do not give a completely accurate portrayal of the behavior of solutions. They do provide a good picture of the qualitative or long term behavior of solutions.

The four images are the slope (or direction) field, the ty-graphs of solutions, the phase line, and the graph of f as a function of y. Of course, the slope field and solution graphs are standard topics in most differential equations courses. Since we are dealing with autonomous equations, we can capture all of the information provided by the slope field in a simpler, one-dimensional picture. In this picture, called the phase line, we record the locations of all of the equilibrium points of f (the zeroes of f(y)) as well as the direction of solution curves y(t). Since the slope field generated by f(y) is independent of t, this is all the information we need to understand the fate of solutions. In Figure 1 we sketch the slopefield, solution graphs, and phase line for

dy / dt = y3 - y.

Figure 1: The slope field, phase line, and solution graphs

In the phase line, circles indicate rest points, while arrows indicate the directions that solutions move as t increases. The existence and uniqueness theorem guarantees that solution curves do not cross the equilibrium points. Consequently, a phase line as depicted in Figure 2 implies that solutions between the equilibrium points p and q tend from q to p as t runs from plus to minus infinity.

Figure 2: Solutions tend from q to p as time increases

Two quick comments are in order. One has to argue that solution curves in Figure 2 actually do tend to p as t tends to infinity (and to q as t tends to minus infinity), but this is a relatively straightforward argument using the slope field. Also, these qualitative methods cannot tell us when a solution tends to infinity in finite time, but they do allow us to conclude that the solution is unbounded.

One important reason for introducing the phase line early is the fact that much of our emphasis later in the course is on systems of equations rather than higher order equations. This means that students need to be able to understand parametrized curves in the xy-plane and to interpret the behavior of the x(t) and y(t) components of the solution. The phase line gives the students early practice doing this which serves them well in this later activity.

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Robert L. Devaney
May 6 1995