Chapter 1 - Introduction to Differential Equations

Abstract

This chapter provides an overview of differential equations. A differential equation is an equation that contains the derivative or differentials of one or more dependent variables in regard to one or more independent variables. If the equation contains only ordinary derivatives of one or more independent variables in regard to a single independent variable, the equation is called an ordinary differential equation. The chapter illustrates ordinary differential equations. Differential equations can be categorized into groups of equations that may be solved in similar ways. The chapter discusses first level of classification, distinguishing ordinary and partial differential equations. It extends this classification system with the definition that the highest derivative in the differential equation is called the order of the equation. When faced with a differential equation, the goal is to determine solutions to the equation. A solution of a differential equation on a given interval is a function that is continuous on the interval and has all the necessary derivatives that are present in the differential equation such that when substituted into the equation yields an identity for all values on the interval. The chapter further reviews initial and boundary value problems. As first-order equations involve a single auxiliary condition, which is usually referred to as an initial condition, the chapter distinguishes between the initial and boundary value problems through some examples.