MTH 253 Calculus (Other Topics)First-Order DE ApplicationsExample 1: Population GrowthSlide 4Other Exponential Growth/Decay ApplicationsMTH 253Calculus (Other Topics)Chapter 9 – Mathematical Modeling with Differential EquationsSection 9.3 – Modeling with First-Order Differential EquationsCopyright © 2006 by Ron Wallace, all rights reserved.First-Order DE ApplicationsSituation provides a known rate of change.Dependent on a function and/or an independent variable (usually t = time)Requires an initial condition (IVP)Solution finds the function whose rate of change is given.0( , ) ( )odyf t y y t ydt= =Find:( )y tExample 1: Population GrowthAssumption:The rate of change of a population is directionally proportional to the current size of the population.0 (0)dyky y ydt= =( ) Population at time y t t=Time StartsBeginning Known PopulationConstant of Proportionalitydykdty=dykdty=� �Example 1: Population GrowthSolution:0 (0)dyky y ydt= =ln y kt c= +( ) ckts ey se==0kty y e=Exponential Growth (k > 0) or Decay (k < 0).Other Exponential Growth/Decay ApplicationsPharmacologyThe amount of a drug in a body changes at a rate inversely proportional to the amount of the drug present.Radio ActivityThe amount of a radioactive material changes at a rate inversely proportional to the amount of the material present.In these two cases, k < 0
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