Overview

Loosely speaking, a differential equation is an equation specifying a relation between the derivatives of a function or between one or more derivatives and the function itself. We will call the function appearing in such an equation the unknown function. We use this terminology because the typical task involving a differential equation, and the focus of this book, is to solve the differential equation, that is, to find a function whose derivatives are related as specified by the differential equation. In carrying out this task, everything else about the relation, other than the unknown function, is regarded as known. Any function satisfying the differential equation is called a solution. In other words, a solution of a differential equation is a function that, when substituted for the unknown function, causes the equation to be satisfied.

Differential equations fall into several natural and widely used categories:

1. Ordinary versus partial:

   1. If the unknown function has a single independent variable, say t, then the equation is an ordinary differential equation (ODE). In this case, only "ordinary" derivatives are involved. Examples of ODEs are

      

      and

      

      In the second example, a, b, and c are regarded as known constants, and f ( t) as a known function of t. In both equations, the unknown is u = u( t).

   2. If the unknown function has two or more independent variables, the equation is called a partial differential equation (PDE). Examples include