Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

The logistic equation for the population (in thousands) of a certain species is given by $\frac{dp}{dt} = 3 p - 2 p^{2}$ .
⦁ Sketch the direction field by using either a computer software package or the method of isoclines.
⦁ If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?
⦁ If $p (0) = 0 . 8$ , what is $\lim_{t \to + \infty} p (t)$ ?
⦁ Can a population of 2000 ever decline to 800?

⦁ The Sketch is drawn for the direction field

⦁ The limiting population is $\frac{3}{2}$

⦁ No

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

1(a): Drawing the Sketch for the direction field of the given equation

Hence, the Sketch is drawn for the direction field.

3(b): Applying the initial condition p(0)=3

Hence, the limiting population is .

4(c): Applying the initial condition p(0)=0.8 in the solution

$\frac{3}{2 - 2 c_{2}} = 0.8 c_{2} = 1 - \frac{3}{1.6} c_{2} = - 0.875 Now, p = \frac{3 e^{3 t}}{2 e^{3 t} + 1.75} \lim_{t \to \infty} p (t) = \frac{3}{2}$

Hence, the limiting population is $\frac{3}{2}$ .

5(d): Analyzing the graph and the different initial conditions

From the above two parts (b), (c) and the graph,

the limiting value of population approaches 1.5 (i.e., 1500) as t tends to infinity.

Hence, the population of 2000 can never decline to 800.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Step by Step Solution

1(a): Drawing the Sketch for the direction field of the given equation

3(b): Applying the initial condition p(0)=3

4(c): Applying the initial condition p(0)=0.8 in the solution

5(d): Analyzing the graph and the different initial conditions

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Step by Step Solution

1(a): Drawing the Sketch for the direction field of the given equation

3(b): Applying the initial condition p(0)=3

4(c): Applying the initial condition p(0)=0.8 in the solution

5(d): Analyzing the graph and the different initial conditions

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