Chapter 4 Differential Equations

1 DVI file created at 14:20, 21 May 2008 Copyright 1994, 2008 Five Colleges, 4 Differential EquationsThe rate Equations with which we began our study of calculus are calleddifferential equationswhen we identify the rates of change that appearwithin them as derivatives of functions. Differential Equations are essentialtools in many area of mathematics and the sciences. In this Chapter weexplore three of their important uses: Modellingproblems using differential Equations ; Solvingdifferential Equations , both through numerical techniqueslikeEuler s method and, where possible, through finding formulas whichmake the Equations true; Definingnew functions by differential also introduce two important functions theexponential functionandthelogarithmic function which play central roles in the theory of solvingdifferential Equations .

2 Finally, we introduce the operation ofantidifferen-tiationas an important tool for solving some special kinds of Modelling with Differential EquationsTo analyze the way an infectious disease spreads through a population, weasked how three quantitiesS,I, andRwould vary over time. This wasdifficult to answer; we found no simple, direct relation betweenS(orIorR) andt. What wedidfind, though, was a relation between the variables179 DVI file created at 14:20, 21 May 2008 Copyright 1994, 2008 Five Colleges, 4. DIFFERENTIAL EQUATIONSS,I, andRand their ratesS ,I , andR . We expressed the relation as aset of rate Equations . Then, given the rate Equations and initial values forS,I, andR, we used Euler s method to estimate the values at any time in thefuture. By constructing a sequence of successive approximations, we wereable to make these estimates as accurate as we are two ideas here.

3 The first is that we could write down equationsfor the rates of change that reflected important features of the process wesought to model. The second is that these equationsdeterminedthe variablesas functions of time, so we could make predictions about the real process wewere modelling. Can we apply these ideas to other processes?To answer this question, it will be helpful to introduce somenew equationsand initial valueproblemsWhat we have been calling rate Equations are more commonly calleddif-ferential Equations . (The name is something of an historical the Equations involve functions and their derivatives, we might bet-ter call themderivativeequations.) Euler s method treats the differentialequations for a set of variables as a prescription for findingfuture values ofthose variables.

4 However, in order to get started, we must always specifythe initial values of the variables their values at some given time. We callthis specification aninitial condition. The differential Equations togetherwith an initial condition is called aninitial value problem. Each initialvalue problem determines a set of functions which we find by using Euler we use Leibniz s notation for derivatives, a differentialequation likeS = aSItakes theformdS/dt= aSI. If we then treatdS/dtas a quotient of the individualdifferentialsdSanddt(see page 123), we can even write the equation asdS= aSI dt. Since this expressesthe differentialdSin terms of the differentialdt, it was natural to call it a differential approach is similar to Leibniz s, except that we don t need to introduce infinitesimally smallquantities, which differentials were for Leibniz.

5 Instead,we write S aSI tand rely on thefact that the accumulated error of the resulting approximations can be made as small as we illustrate how differential Equations can be used to describe a widerange of processes in the physical, biological, and social sciences, we ll devotethis section to a number of ways to model and analyze the long-term behaviorof animal populations. To be specific, we will talk about rabbits and foxes,but the ideas can be adapted to the population dynamics of virtually all livingthings (and many non-living systems as well, such as chemical reactions).In each model, we will begin by identifying variables that describe whatis happening. Then, we will try to establish how those variables change overtime. Of course, no model can hope to capture every feature ofthe pro-DVI file created at 14:20, 21 May 2008 Copyright 1994, 2008 Five Colleges, MODELLING WITH DIFFERENTIAL EQUATIONS181cess we seek to describe, so we begin simply.

6 We choose just one or twoelements that seem particularly important. After examining the predictionsof our simple model and checking how well they correspond to reality, wemake modifications. We might include more features of the population dy-Models canprovide successiveapproximationsto realitynamics, or we might describe the same features in different ways. Gradually,through a succession of refinements of our original simple model, we hope fordescriptions that come closer and closer to the real situation we are Models: RabbitsThe problem. If we turn 2000 rabbits loose on a large, unpopulated islandthat has plenty of food for the rabbits, how might the number of rabbits varyover time? If we letR=R(t) be the number of rabbits at timet(measured inmonths, let us say), we would like to be able to make some predictions aboutthe functionR(t).

7 It would be ideal to have a formula forR(t) but this isnot usually possible. Nevertheless, there may still be a great deal we can sayabout the behavior ofR. To begin our explorations we will construct a modelof the rabbit population that is obviously too simple. Afterwe analyze thepredictions it makes, we ll look at various ways to modify the model so thatit approximates reality more first model. Let s assume that, at any timet, the rate at which theConstantper capita growthrabbit population changes is simply proportional to the number of rabbitspresent at that time. For instance, if there were twice as many rabbits, thentherateat which new rabbits appear will also double. In mathematicalterms, our assumption takes the form of the differential equation(1)dRdt=k multiplierkis called theper capita growth rate(or thereproductiverate), and its units are rabbits per month per rabbit.

8 Per capita growth isdiscussed in exercise 22 in Chapter 1, section the sake of discussion, let s suppose thatk=.1 rabbits per month perrabbit. This assumption means that, on the average, one rabbit will new rabbits every month. In theS-I-Rmodel of Chapter 1, the reciprocalsof the coefficients in the differential Equations had natural same is true here for the per capita growth rate. Specifically, we can saythat 1/k= 10 months is the average length of time required for a rabbittoproduce one new file created at 14:20, 21 May 2008 Copyright 1994, 2008 Five Colleges, 4. DIFFERENTIAL EQUATIONSS ince there are 2000 rabbits at the start, we can now state a clearlydefined initial value problem for the functionR(t):dRdt=.1RR(0) = modifying the program SIRPLOT, we can readily produce thegraph ofUse Euler s methodto findR(t)the function that is determined by this problem.

9 Before we dothat, though,let s first consider some of the implications that we can drawout of theproblem without the (t) =.1R(t) rabbits per month andR(0) = 2000 rabbits, we seethat the initial rate of growth isR (0) = 200 rabbits per month. If this ratewere to persist for 20 years (= 240 months),Rwould have increased by R= 240 months 200rabbitsmonth= 48000 rabbits,yielding altogetherR(240) =R(0) + R= 2000 + 48000 = 50000 rabbitsat the end of the 20 years. However, since the populationRis always gettinglarger, the differential equation tells us that the growth rateR willalsoalways be getting larger. Consequently, 50,000 is actuallyan underestimateof the number of rabbits predicted by this s restate our conclusions in a graphical form. IfR were always 200rabbits per month, the graph ofRplotted againsttwould just be a straightline whose slope is 200 rabbits/month.

10 ButR is always getting bigger, soThe graph ofRcurves upthe slope of the graph should increase from left to right. This will make thegraph curve upward. In fact, SIRPLOT will produce the following graph ofR(t):20000number of rabbitsgraph if rabbitsincreased at 200per month foreveractual graphtDVI file created at 14:20, 21 May 2008 Copyright 1994, 2008 Five Colleges, MODELLING WITH DIFFERENTIAL EQUATIONS183 Later, we will see that the functionR(t) determined by this initial valueproblem is actually an exponential function oft, and we will even be able towrite down a formula forR(t), namelyR(t) = 2000 ( ) model is too simple to be able to describe what happens toa rabbitpopulation very well. One of the obvious difficulties is that it predicts therabbit population just keeps growing forever.