# CHAPTER 1

# First-Order ODEs

Chapter 1 begins the study of ordinary differential equations (ODEs) by deriving them from physical or other problems (*modeling*), solving them by standard mathematical methods, and interpreting solutions and their graphs in terms of a given problem. The simplest ODEs to be discussed are ODEs *of the first order* because they involve only the first derivative of the unknown function and no higher derivatives. These unknown functions will usually be denoted by *y*(*x*) or *y*(*t*)when the independent variable denotes time *t*. The chapter ends with a study of the existence and uniqueness of solutions of ODEs in Sec. 1.7.

Understanding the basics of ODEs requires solving problems by hand (paper and pencil, or typing on your computer, but first without the aid of a CAS). In doing so, you will gain an important conceptual understanding and feel for the basic terms, such as ODEs, direction field, and initial value problem. If you wish, you can use your **Computer Algebra System (CAS)** for checking solutions.

**COMMENT. ***Numerics for first-order ODEs can be studied immediately after this chapter.* See Secs. 21.1–21.2, which are independent of other sections on numerics.

*Prerequisite:* Integral calculus.

*Sections that may be omitted in a shorter course:* 1.6, 1.7.

*References and Answers to Problems:* App. 1 Part A, and App. 2.

## 1.1 Basic Concepts. Modeling

If we want to solve an engineering ...