Transcription of Chapter 4 Differential Equations
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Chapter 4 Differential Equations
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 . 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.
ferential equations. (The name is something of an historical accident. Since the equations involve functions and their derivatives, we might bet-ter call them derivative equations.) Euler’s method treats the differential equations for a set of variables as a prescription for finding future values of those variables.
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