Economics Dynamics Problems 49

Economics Dynamics Problems 49 - Example 2.7 In many...

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Continuous dynamic systems 33 Figure 2.4. Example 2.6 Assume population, p , grows at a constant rate k , where we assume that p is a continuous function of time, t . This means that the percentage change in the population is a constant k . Hence dp dt 1 p = k (2.8) which immediately gives the Frst-order linear homogeneous differential equation dp dt kp = 0 with solution p ( t ) = p 0 e kt which satisFes the initial condition p (0) = p 0 . Typical solution paths for this Malthusian population growth are illustrated in Fgure 2.4.
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Unformatted text preview: Example 2.7 In many scientiFc problems use is made of radioactive decay. Certain radioactive elements are unstable and within a certain period the atoms degenerate to form another element. However, in a speciFed time period the decay is quite speciFc. In the early twentieth century the famous physicist Ernest Rutherford showed that the radioactivity of a substance is directly proportional to the number of atoms...
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