Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 10 - Introduction to Differential Equations - 10.3 The Logistic Equation - Exercises - Page 518: 2


$$y= \frac{3}{1-7e^{-6t}/10}.$$

Work Step by Step

Rewriting the logistic equation as $\frac{d y}{d t}=6 y\left(1-\frac{y}{3}\right)$, we have $k=6$ and $A=3$ and hence the solution is given by $$y=\frac{A}{1-e^{-kt}/B}=\frac{3}{1-e^{-6t}/B}.$$ Since $y(0)=10$, then $10=\frac{3}{1-1/B}$ and hence $10=\frac{3B}{B-1}$. Then $10B-10=3B$ and $B=\frac{10}{7}$ and $$y= \frac{3}{1-7e^{-6t}/10}.$$
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