## ABSTRACT

It turns out that first-order differential equations can often be solved or otherwise analyzed using methods developed directly from the tools developed in elementary calculus. This chapter is the first of several concentrating on first-order equations and these calculus-based methods. This chapter introduces the “derivative formula form” for first order differential equations, the definition of an autonomous differential equation, and the notion of “constant” or “equilibrium” solutions (which will later be found particularly important in determining the long-term behavior of nonconstant solutions). Also in this chapter are the fundamental theorems on the existence and uniqueness of solutions to first-order initial-value problems. These theorems are proven. The first proof starts with an easy-to-follow outline of the proof using the Picard sequence. This outline is then followed by a rigorous discussion of the technical details for the more advanced reader. Finally, there is an appendix on multivariable calculus for those readers who have not yet been formally introduced to the calculus of functions of two or more variables (e.g., limits and partial derivatives of multivariable functions).