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: 8

Answer

$\frac{dy}{dt}=-3y+2$

Work Step by Step

We must write a differential equation of the form $\frac{dy}{dt}=ay+b$ such that all solutions approach $y=\frac{2}{3}$ as $t\rightarrow\infty$. Solution: We want to have the solution function to have slope $0$ when $y=\frac{2}{3}$. Thus, $0=a(\frac{2}{3})+b$. So, $\frac{b}{a}=-\frac{2}{3}$. Hence, one such equation satisfying this criteria is $\frac{dy}{dt}=-3y+2$. This may be rewritten $y'=2-3y$.
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