Separable D.E.’s -
Growth and Decay Applications
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In a Nut Shell: There are several mathematical models to represent growth and decay. Two of the simpler ones are given below.
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Differential equation involving “growth” dy/dt = K y ; here rate of change of y is proportional to itself subject to y(0) = yo which represents the initial condition of y at t = 0 K = proportionality constant Applications: a.
population growth b. annual rate (income) dy/y = K dt (separate variables y and t and integrate) ln y = K t + C1 , C1 is the constant of integration y = e Kt
+ C1 = C e Kt |
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Differential equation involving
“decay” (negative growth) dy/dt = - K y ; here rate of change of y is proportional to its subject to y(0) = yo which represents the initial condition of y at t = 0 K = proportionality constant i.e. K = decay constant K = sales decay constant K = drug elimination constant Applications: a.
radioactive decay (radiocarbon
dating) b. annual rate (income) dy/y = - K dt (separate variables y and t and integrate) ln y = - K t + C1 , C1 is the constant of integration y = e
-Kt + C1 = C e –Kt |
Copyright © 2011 Richard C. Coddington