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Unformatted text preview: August 30, 2011 4-1 4. Some Applications of first order linear dif- ferential Equations The modeling problem There are several steps required for modeling scientific phenomena 1. Data collection (experimentation) Given a certain physical system, one has to run experiments and get some idea of how the observed data depend on time. 2. Setting up scientific law to describe the time dependence This may involve differential or difference equations. The idea is to find the correct equations whose solutions give the observed time de- pendence. 3. Analysis of solutions of appropriate equations to describe observed phe- nomena. We will describe several known applications involving this process. Radioactive Decay It is known that certain radioactive substances exhibit spontaneous decay. That is, if Q ( t ) represents the amount of the substance at time t , then Q ( t ) satisfies the differential equation dQ dt =- rQ ( t ) (1) where r is a positive real number. This simply means that the rate of decay of the quantity at time t is proportional to the amount present at time t . We know that the general solution to (1) is Q ( t ) = Q (0) e- rt where Q (0) is the amount present at time 0. We can use this to solve various questions related to radio-active decay. 1. The element Thorium-234 (Th-234) exhibits radio-active decay. If 100 mg of Th-234 decays to 82.04 mg in one week, find an expression for August 30, 2011 4-2 the amount at any time t . Also, find the half-life of the element (the amount of time it takes to decay to half its original value). Let Q ( t ) denote the amount at time t . Let Q = Q (0). Then, Q ( t ) = Q e- rt . If t is measured in units of days, and Q ( t ) is measured in units of milligrams (mg), then Q = 100 , Q (7) = 100 e- 7 r = 82 . 04 , e- 7 r = 82 . 04 / 100 , r =- log (82 . 04 / 100) 7 = 0 . 028 ....
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