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### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# The logistic equation for the population (in thousands) of a certain species is given by $\frac{\mathbf{dp}}{\mathbf{dt}}{\mathbf{=}}{\mathbf{3}}{\mathbf{p}}{\mathbf{-}}{\mathbf{2}}{{\mathbf{p}}}^{{\mathbf{2}}}$ .⦁ 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 $\mathbf{p}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}{\mathbf{.}}{\mathbf{8}}$ , what is ${{\mathbf{lim}}}_{\mathbf{t}\to \mathbf{+}\infty }\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ ?⦁ Can a population of 2000 ever decline to 800?

⦁ The Sketch is drawn for the direction field

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

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

⦁ No

See the step by step solution

## 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

$\frac{\mathbf{3}}{\mathbf{2}\mathbf{-}\mathbf{2}{\mathbf{c}}_{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}\phantom{\rule{0ex}{0ex}}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{3}}{\mathbf{1}\mathbf{.}\mathbf{6}}\phantom{\rule{0ex}{0ex}}{\mathbf{c}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{875}\phantom{\rule{0ex}{0ex}}\mathbf{Now}\mathbf{,}\mathbf{p}\mathbf{=}\frac{\mathbf{3}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}}{\mathbf{2}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{75}}\phantom{\rule{0ex}{0ex}}{\mathbf{lim}}_{t\to \infty }\mathbf{p}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}$

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

## 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.