Summary of Chapter 1 : Basic Concepts

We will develop ways of analyzing ordinary differential equations that take full advantage of the power of calculus and technology. We do this by treating topics from graphical, numerical, and analytical points of view. Because each of these points of view can at times give incomplete information, we always need to compare our results for consistency. The strategies presented in this book will empower you to correctly analyze solutions of a differential equation, even when those solutions are not obtainable as analytical expressions.

In this chapter, we introduce these three points of view by considering differential equations of the form dy/dx=g(x). We start with a brief introduction, definitions, and examples, which make use of prior knowledge of antiderivatives. Then we spend the next two sections illustrating graphical techniques, including slope fields and isoclines. These techniques often let us discover many properties of the solution of a specific differential equation by simply analyzing the differential equation from a graphical point of view. We end this chapter with a discussion of Taylor series, which are especially useful when solutions are given by integrals that do not have simple antiderivatives.

The purpose of this chapter is to illustrate graphical, numerical, and analytical approaches in the familiar setting of antiderivatives. We want to make sure you have a firm foundation in these approaches so you can quickly grasp the new ideas in subsequent chapters.

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