Explanation Given: The given values are a = 5 , b = 1 and h = 25 4 . Calculation: Consider the differential equation for the model of the population P ( t ) . d P d t = P ( a − b P ) − h (1) Substitute 5 for a , 1 for b and 25 4 for h in Equation (1). d P d t = P ( 5 − ( 1 ) P ) − 25 4 = P ( 5 − P ) − 25 4 (2) Use quadratic formula to solve for P . P = 5 ± 5 2 − 4 ( 1 ) ( 25 4 ) 2 ( 1 ) = 5 2 Substitute 1 for P in Equation (2). d P d t = 1 ( 5 − 1 ) − 25 4 = 4 − 25 4 = − 9 4 Substitute 4 for P in Equation (2). d P d t = 4 ( 5 − 4 ) − 25 4 = 4 − 25 4 = − 9 4 The following figure shows the phase diagram and the solution curve. Figure (1) Consider the differential equation. d P d t = P ( 5 − P ) − 25 4 Separate the variables and integrate. d P d t = − P 2 + 5 P − 25 4 d P d t = − ( P − 5 2 ) 2 ∫ d P ( P − 5 2 ) 2 = − ∫ d t − ( P − 5 2 ) − 1 = − t + c Simplify the above equation

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Differential Equations

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Chapter 2.8, Problem 6E

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Whether the population becomes extinct in finite time.

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Chapter 2 Solutions

EBK ADVANCED ENGINEERING MATHEMATICS

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Whether the population becomes extinct in finite time.

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Chapter 2 Solutions