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: 7

Answer

$\frac{dy}{dt}=3-y$.

Work Step by Step

We must write an equation of the form$\frac{dy}{dt} = ay+b$ such that all solutions approach $y=3$ as $t\rightarrow\infty$. Solution: We recognize that at $y=3$, the slope of the function $ay+b$ must be $0$: $0=a(3)+b$, so, $3=-\frac{b}{a}$. In other words, the ratio $\frac{b}{a}$ must be $-3$. So a differential equation of the required form is $\frac{dy}{dt}=-y+3$, or $\frac{dy}{dt}=3-y$.
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