Differential Equations

This and the next three chapters are resources, designed to get the neophyte started, to recall and extend the skills of the initiated, and to call the expert’s attention to the relative importance of what is already known. In this chapter we first introduce and motivate the notation that we use throughout the book, and specifically the operator notation Dx for the derivative of a function x . That is, we view a derivative as a transformation of a function into a new function whose value at any location t is the slope of x at that point. Differential equations are divided into various classes, and we first describe the autonomous/forced dichotomy defined by whether or not the system has an external or exogenous input. Then we consider linear versus nonlinear systems, a distinction that is already familiar to statistical readers in regression analysis. The data are assumed to be distributed over the whole interval of observation, and to be subject to measurement error to other types of random disturbances. This contrasts with the errorless initial and boundary value data configurations that apply to so many models.