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HARVARD MATH Xb - Lab 10: Differential Equations

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Math Xb Spring 2004 NameLab 10: Differential Equations Section InstructorMay 4, 2004 CollaboratorsIn a laboratory, a colony of fruit flies are under scrutiny. Let P = P (t) be the number of fruit flies in thecolony at time t.Given ample food, the population would grow at a rate proportional to itself according todPdt= kP . Butflies are continually being siphoned off to another lab at a constant rate of C flies per day.The rate of change of the fly population is the rate at which it is increasing due to repro duction minus therate at which flies are being siphoned off.We mo del the s ituation as follows.(Rate of change) = (Rate of increase) − (Rate of decrease)dPdt= kP − CWe want to solve this differential equation for P (t).1. Suppose k = 0.02 and C = 2. Then we want to solve the differential equationdPdt= 0.02P − 2.First, rewrite it asdPdt= 0.02(P − 100).We can convert this differential equation to a form with which we are familiar by making the substi-tution y = P − 100.(a) Express the differential equationdPdt= 0.02(P − 100) in terms of y.(b) Solve the differential equation you found in part (a).(c) Knowing that y = P − 100, find P (t). There will be an arbitrary constant in your answer becauseyou will have found the general solution to the differential equation.(d) Suppose that P (0) = 3000. Find the particular solution corresponding to this initial condition.2. Use similar substitution techniques to solve the following differential equations.(a)dQdt= 2Q − 6(b)dMdt= 0.1M −


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HARVARD MATH Xb - Lab 10: Differential Equations

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