(a) Explanation Given: The given conditions are P 0 > e and 0 < P 0 < e . Calculation: Consider the Gompertz differential equation. d P d t = P ( a − b ln P ) (1) Substitute 1 for a and b in Equation (1). d P d t = P ( 1 − ln P ) (2) To create phase portrait for the differential equation, find equilibrium solution. d P d t = 0 P ( 1 − ln P ) = 0 Here, ln e = 1 . Therefore, P = 0 and P = e and the two intervals are 0 < P < e and e < P (b) To determine To sketch: The solution curves corresponding to the cases P 0 > e − 1 and 0 < P 0 < e − 1 . (c) To determine An explicit solution of the Gompertz differential equation.

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.8, Problem 8E

(a)

To determine

To sketch: The solution curves corresponding to the cases P0>e and 0<P0<e.

(b)

To determine

To sketch: The solution curves corresponding to the cases P0>e−1 and 0<P0<e−1.

(c)

To determine

An explicit solution of the Gompertz differential equation.

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Chapter 2.8, Problem 7E

Chapter 2.8, Problem 9E

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Chapter 2 Solutions

Advanced Engineering Mathematics

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