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Ordinary Diff Eq Exam Review Solutions 1

Ordinary Diff Eq Exam Review Solutions 1 - 1 Solutions...

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1 Solutions Section 1.1 1. The rate of change in the population P ( t ) is the derivative P 0 ( t ) . The Malthusian Growth Law states that the rate of change in the population is proportional to P ( t ) . Thus P 0 ( t ) = kP ( t ) , where k is the proportionality constant. Without reference to the t variable, the differential equation becomes P 0 = kP 2. a. This statement mathematically is b ( t ) = b 0 P ( t ) where we have used b 0 to represent the proportionality constant. b. This statement translates as d ( t ) = d 0 P 2 ( t ) where we have used d 0 to represent the proportionality constant. c. The overall growth rate is P 0 ( t ) . Thus the Logistic Growth Law is P 0 ( t ) = b ( t ) d ( t ) = b 0 P ( t ) d 0 P 2 ( t ) = ( b 0 d 0 P ( t )) P ( t ) : 3. Torricelli’s law states that the change in height,
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