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Digital Signal Processing Handbook Part 2

A function containing variables and their derivatives is called a differential expression, and an equation involving differential expressions is called a differential equation. A differential equation is an ordinary differential equation if it contains only one independent variable; it is a partial differential equation if it contains more than one independent variable. We shall deal here only with ordinary differential equations. In the mathematical texts, the independent variable is generally x, which can be anything such as time, distance, velocity, pressure, and so on. In most of the applications in control systems, the independent variable is time | Ordinary Linear Differential and Difference Equations B.P. Lathi California State University Sacramento 2.1 Differential Equations Classical Solution Method of Convolution 2.2 Difference Equations Initial Conditions and Iterative Solution Classical Solution Method of Convolution References 2.1 Differential Equations A function containing variables and their derivatives is called a differential expression and an equation involving differential expressions is called a differential equation. A differential equation is an ordinary differential equation if it contains only one independent variable it is a partial differential equation if it contains more than one independent variable. We shall deal here only with ordinary differential equations. In the mathematical texts the independent variable is generally x which can be anything such as time distance velocity pressure and so on. In most of the applications in control systems the independent variable is time. For this reason we shall use here independent variable t for time although it can stand for any other variable as well. The following equation yy 3 y 5y2.t sin t dt2 dt is an ordinary differential equation of second order because the highest derivative is of the second order. An nth-order differential equation is linear if it is of the form dny dn-1y dy an t dtn an-1 t dtn-1 a1 t dt ao t y t r t 2.1 where the coefficients ai t are not functions of y t . If these coefficients a are constants the equation is linear with constant coefficients. Many engineering as well as nonengineering systems can be modeled by these equations. Systems modeled by these equations are known as linear timeinvariant LTI systems. In this chapter we shall deal exclusively with linear differential equations with constant coefficients. Certain other forms of differential equations are dealt with elsewhere in this volume. 1999 by CRC Press LLC Role of Auxiliary Conditions in Solution of Differential Equations We now show that a differential