In the past several chapters, the efforts were mainly directed toward finding formulas or equations describing solutions to given differential equations. Then, sometimes, the graphs of these solutions were sketched using those formulas and equations. In this chapter, something quite different is done. Instead of being solved, the differential equations are used, directly, to sketch the graphs of their solutions. No other formulas or equations are needed.

The graphic techniques and underlying ideas that are developed in this chapter are, naturally, especially useful when dealing with first-order differential equations that cannot be solved using the methods developed in previous chapters. But these methods can be valuable even when the given differential equation can be solved because they yield “pictures” describing the general behavior of the possible solutions. Often, these pictures are even more enlightening that the formulas for the solutions.

In particular, these pictures lead naturally to a discussion of “stability” in differential equations, and to an expansion of the discussion of the existence and uniqueness of solutions, along with clear illustrations of when existence or uniqueness may fail to occur.