Tài liệu 02 Ordinary Linear Differential and Difference Equations docx

Lathi, B.P. “Ordinary Linear Differential and Difference Equations”

Digital Signal Processing Handbook

Ed. Vijay K. Madisetti and Douglas B. Williams

Boca Raton: CRC Press LLC, 1999

c 1999 by CRC Press LLC

2

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

d2y

dt2

4

+ 3

dy

dt + 5y2(t) = sin t

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

an(t) dny

dtn + an−1(t) dn−1y

dtn−1 +···+ a1(t) dy

dt

+ a0(t)y(t) = r(t) (2.1)

where the coefficients ai(t) are not functions of y(t). If these coefficients (ai) 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 time￾invariant (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.

c 1999 by CRC Press LLC