Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 1 - Introduction - 1.1 Some Basic Mathematical Models; Direction Fields - Problems - Page 8: 17


(g) $\frac{dy}{dt}=-2-y$

Work Step by Step

The directional field for question 17 shows a field with an equilibrium solution at y=-2; it converges as $t →∞$. Because there is only one equilibrium solution, it can be inferred that the differential equation is of the form $\frac{dy}{dt}=(a+by)^n$, this is because only equations in this form can have a single equilibrium solution. This rules out answers (d), (e), and (h). All remaining answer choices are of the form $\frac{dy}{dt}=a+by$ . The equation is convergent, which means that $b\lt0$. This rules out answers (a), (b), (c), and (f). The differential equation has an equilibrium solution ($\frac{dy}{dt}=0$) at $y=-2$, therefore $-\frac{a}{b}=-2$. So, $a=2b$. The equation is of the form $\frac{dy}{dt}=2b+by$, and b is negative. Therefore the answer is (g).
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