1 INTRODUCTION TO DIFFERENTIAL EQUATIONS

1 11 INTRODUCTION TO and EQUATIONS as Mathematical ModelsCHAPTER 1 IN REVIEWThe words differentialandequationscertainly suggest solving some kind ofequation that contains derivatives y ,y , .. Analogous to a course in algebra andtrigonometry, in which a good amount of time is spent solving EQUATIONS such asx2 5x 4 0 for the unknown number x, in this course oneof our tasks will beto solve DIFFERENTIAL EQUATIONS such as y 2y y 0 for an unknown functiony (x).The preceding paragraph tells something, but not the complete story, about thecourse you are about to begin.

2 As the course unfolds, you will see that there is moreto the study of DIFFERENTIAL EQUATIONS than just mastering methods that someone hasdevised to solve first things first. In order to read, study, and be conversant in a specializedsubject, you have to learn the terminology of that discipline. This is the thrust of thefirst two sections of this chapter. In the last section we briefly examine the linkbetween DIFFERENTIAL EQUATIONS and the real world. Practical questions such as Howfast does a disease spread?

3 How fast does a population change?involve rates ofchange or derivatives. As so the mathematical description or mathematicalmodel of experiments, observations, or theories may be a DIFFERENTIAL AND TERMINOLOGYREVIEW MATERIAL Definition of the derivative Rules of differentiation Derivative as a rate of change First derivative and increasing/decreasing Second derivative and concavityINTRODUCTIONThe derivative dy dxof a function y (x) is itself another function (x)found by an appropriate rule. The function is differentiable on the interval ( , ), andby the Chain Rule its derivative is.

4 If we replace on the right-hand side ofthe last equation by the symbol y, the derivative becomes.(1)Now imagine that a friend of yours simply hands you equation (1) you have no idea how it wasconstructed and asks, What is the function represented by the symbol y? You are now face to facewith one of the basic problems in this course: How do you solve such an equation for the unknown function y (x)?dydx >dx CHAPTER 1 INTRODUCTION TO DIFFERENTIAL DEFINITIONThe equation that we made up in (1) is called a proceeding any further, let us consider a more precise definition ofthis EquationAn equation containing the derivatives of one or more dependent variables,with respect to one or more independent variables, is said to be a differentialequation (DE).

5 To talk about them, we shall classify DIFFERENTIAL EQUATIONS by type, order, BY TYPEIf an equation contains only ordinary derivatives ofone or more dependent variables with respect to a single independent variable it issaid to be an ordinary DIFFERENTIAL equation (ODE).For example,A DE can contain morethan one dependent variable(2)are ordinary DIFFERENTIAL EQUATIONS . An equation involving partial derivatives ofone or more dependent variables of two or more independent variables is called adydx 5y ex, d2ydx2 dydx 6y 0, and dxdt dydt 2x yb bpartial DIFFERENTIAL equation (PDE).

6 For example,(3)are partial DIFFERENTIAL EQUATIONS .*Throughout this text ordinary derivatives will be written by using either theLeibniz notationdy dx,d2y dx2,d3y dx3,.. or theprime notationy ,y ,y ,..By using the latter notation, the first two DIFFERENTIAL EQUATIONS in (2) can be writtena little more compactly as y 5y exandy y 6y 0. Actually, the primenotation is used to denote only the first three derivatives; the fourth derivative iswritteny(4)instead of y . In general, the nth derivative of yis written dny dxnory(n).

7 Although less convenient to write and to typeset, the Leibniz notation has an advan-tage over the prime notation in that it clearly displays both the dependent andindependent variables. For example, in the equationit is immediately seen that the symbol xnow represents a dependent variable,whereas the independent variable is should also be aware that in physicalsciences and engineering, Newton s dot notation(derogatively referred to by someas the flyspeck notation) is sometimes used to denote derivatives with respectto time the DIFFERENTIAL equation d2s dt2 32 becomes s 32.

8 Partialderivatives are often denoted by a subscript notationindicating the indepen-dent variables. For example, with the subscript notation the second equation in(3) becomes uxx utt BY ORDERT heorder of a DIFFERENTIAL equation(eitherODE or PDE) is the order of the highest derivative in the equation. For example,is a second-order ordinary DIFFERENTIAL equation. First-order ordinary differentialequations are occasionally written in DIFFERENTIAL form M(x,y)dx N(x,y)dy example, if we assume that ydenotes the dependent variable in (y x)dx 4xdy 0, then y dy dx, so by dividing by the DIFFERENTIAL dx, weget the alternative form 4xy y x.

9 See the Remarksat the end of this symbols we can express an nth-order ordinary DIFFERENTIAL equation in onedependent variable by the general form,(4)whereFis a real-valued function of n 2 variables: x,y,y ,.., y(n). For both prac-tical and theoretical reasons we shall also make the assumption hereafter that it ispossible to solve an ordinary DIFFERENTIAL equation in the form (4) uniquely for theF(x,y,y , .. , y(n)) 0first ordersecond order 5()3 4y exdy dxd2y dx2d2x dt2 16x 0unknown functionor dependent variableindependent variable 2u x2 2u y2 0, 2u x2 2u t2 2 u t, and u y v AND TERMINOLOGY 3*Except for this introductory section, only ordinary DIFFERENTIAL EQUATIONS are considered in A FirstCourse in DIFFERENTIAL EQUATIONS with Modeling Applications,Ninth Edition.

10 In that text thewordequationand the abbreviation DE refer only to ODEs. Partial DIFFERENTIAL EQUATIONS or PDEsare considered in the expanded volume DIFFERENTIAL EQUATIONS with Boundary-Value Problems,Seventh derivative y(n)in terms of the remaining n 1 variables. The differentialequation,(5)wherefis a real-valued continuous function, is referred to as the normal form of (4).Thus when it suits our purposes, we shall use the normal formsto represent general first- and second-order ordinary DIFFERENTIAL EQUATIONS .