1550078481-Ordinary_Differential_Equations__Roberts_

6 Ordinary Differential Equati ons

four roots to this equation and they can all b e found in the set of complex

numbers. This last fact is due to the fundamental theorem of algebra

which st a tes:

"E very polynomial of degree n ::::: 1 with complex coefficients has n

roots (not necessarily distinct) among the complex numbers."

The funda mental theorem of algebra is an existence theorem, since it states

that there exist n roots t o a polynomial equation of degree n. Of course, the

set of roots of a polynomial equation is unique. So the solution of the equation

2x^4 - 3x^3 - 1 3x^2 + 37 x - 1 5 = 0 is a set of four complex numbers and tha t

set is unique. Ca n you solve this equation?

Throughout this t ext, we will state exist ence and uniqueness theorems for

various typ es of problems involving differential equations and syst ems of dif-

ferential equ ations. Before ge nerating a numerical solut io n to any su ch prob-

lem , it is necessary to verify that t he hyp otheses of appropria t e existen ce and

uniqueness theorems are satisfied. Otherwise, the computer may generat e a

"solution" where none exists or it may generate a single solution in a region

where the solut ion is not unique-tha t i s, the computer may generat e a single

solution where there are mult iple solutions. Sometimes we present examples

which illustrat e the erroneo us results one m ay obtain if a numerical solution

is produced without regard for the appropria t e theory.

DEFINITIONS Differential Equation, Independent Variable,

and Dependent Variable

An equation which contains one or more derivative of an unkn own function

or functions or which contains different ials is called a differential equation

(DE).

When a differential equation contains one or more derivatives wit h resp ect

to a p articular vari able, tha t va riable is call ed an independent variable.

A va ri able is said t o b e a dependent variable, if some derivative of the

vari able appears in t he differential equation.

In order t o systematicall y study differential equations, it is convenient and
advantageous to classify the equations into different cat egories. Two broad ,
general cat egories used to classify different ial equations are ordina ry differen-
tial equations and part ial differential equations. This classification is b ased on
t h e type of unknown function appearing in the different ial equation. If the un-
known function dep ends on only one indep endent vari able and the differential

equation contains only ordinary derivatives, then the differential equation is

called a n ordinary differential equation (ODE). If t he unknown function
dep ends on two or more indep endent varia bles and t he different ial equation