This chapter begins with a collection of classic examples of ordinary differential equations (ODEs) from different scientific and technical fields, such as physics and chemistry, engineering, demography, and chaos theory. A short description of these equations is provided to point out their importance in mathematically modeling physical phenomena and real-world situations. Hopefully, these examples will help students realize that ODEs are relevant and important in their respective fields of study. This textbook, as its title suggests, is written for students who have completed Calculus I and II (single-variable calculus). However, after a summer recess, students tend to become rusty at calculus. Acknowledging that reality, basic calculus concepts inextricably tied to ODEs, such as the definitions of the average rate of change and the derivative of a function, are reviewed in this chapter. Techniques of integration are also reviewed in numerous examples, where it is explained in detail how to go about finding solutions of the simplest ODEs, namely, first-order equations of the form \(\displaystyle \frac {dy}{dx} = f(x).\qquad \hbox{(1)} \) Since the right-hand side only involves the independent variable x, solutions of (1) for a given function \(f(x)\) are obtained by finding its indefinite integral. But students are already familiar with this from their Calculus I and II courses. So this is the ideal time to engage and immerse them right away in ODEs by assigning them the problems at the end of this chapter, most of which involve ODEs of the form (1). By solving these problems, students will also be reviewing the basic integration formulas and techniques of integration: substitution, integration by parts, completion of the square, and partial fractions. This will even help those students who have already taken Calculus III (multivariable calculus) since they tend to forget the integration techniques that are part of a Calculus II course. This textbook does not presuppose any familiarity with partial derivatives; however, Chap. 8 covers exact and related differential equations. In preparation for that chapter, first-order partial derivatives are introduced via an example about the temperature variations on the surface of an unevenly heated copper plate. In addition, there are examples and problems concerned with finding solutions of simple partial differential equations of the form \(\displaystyle \frac {\partial {u}}{\partial {x}} = f(x,y) \quad \text{and} \quad \frac {\partial {u}}{\partial {y}} = f(x,y). \qquad \hbox{(2)} \) Simple models and applications of differential equations are introduced in this chapter—not put off until later chapters. For instance, examples and problems of differential equations arising from proportionality statements are presented. An example is the observation that under certain conditions, the rate at which ethane ( \(\text{C}_2 \text{H}_6\) ) decomposes is proportional to its concentration. This observation expressed as an ODE is \(\displaystyle {} \frac {d\left [\text{C}_2 \text{H}_6\right ]}{dt} = -k \left [\text{C}_2 \text{H}_6\right ], \qquad \hbox{(3)} \) where the concentration of ethane is denoted by enclosing its molecular formula in brackets and \(k>0\) is the constant of proportionality. Some other examples involving proportionality assumptions are Newton’s law of cooling and Poiseuille’s law for the volumetric flow rate of a fluid though a cylindrical pipe. Since Newton’s laws of motion are used to introduce a number of different topics in later chapters, a detailed introduction to Newton’s second law of motion is presented so that students who have not had a physics course will have no trouble following these topics. Examples in this chapter include modeling the vertical motion of a rock released from the top of a cliff and a derivation of Tsiolkovsky’s rocket equation. The raison d’être for these models and applications is to motivate students and to incentivize the study of ODEs—and to answer the typical classroom question (whether or not it is actually asked): “What good is all this stuff?”