(a) Explanation Given: The logistic equation is d P d t = k P ( 1 − P A ) , P ( 0 ) = P 0 . Calculation: Consider d P d t = k P ( 1 − P A ) . Separate the variables P and t from the above equation. d P d t = k P ( 1 − P A ) d p k P ( 1 − P A ) = d t Integrate on both side of the equation. ∫ d P P ( 1 − P A ) = ∫ k d t ∫ A d P P ( A − P ) = ∫ k d t ∫ 1 P d P + ∫ 1 A − P d P = ∫ k d t ( ∵ A P ( A − P ) = 1 P + 1 A − P ) ln | P A − P | = k t + D Take exponential on both sides of the above equation. e ln | P A − P | = e k t + D e ln ( P A − P ) = e k t ⋅ e D P ( t ) A − P ( t ) = e k t C (1) Substitute 0 for t in the above equation (b) To determine To Sketch: The graph for the solution curve P ( t ) when k = 0.025 , A = 300 and P 0 = 1 . (c) To determine To describe: The long term behavior of the solution.

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Differential Equations

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Chapter 9.3, Problem 40E

(a)

To determine

To find: The solution of the initial value problem for t>0 and plot the graph for the given solution.

(b)

To determine

To Sketch: The graph for the solution curve P(t) when k=0.025, A=300 and P0=1 .

(c)

To determine

To describe: The long term behavior of the solution.

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Chapter 9 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - The solution to the initial value problem y(t) = 2...

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T o find: The solution of the initial value problem for t > 0 and plot the graph for the given solution.

Solution Summary: The author plots the graph for the given solution of the initial value problem for t>0.

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To find: The solution of the initial value problem for t>0 and plot the graph for the given solution.

To Sketch: The graph for the solution curve P(t) when k=0.025, A=300 and P0=1 .

To describe: The long term behavior of the solution.

Chapter 9 Solutions

T o find: The solution of the initial value problem for t &gt; 0 and plot the graph for the given solution.

Solution Summary: The author plots the graph for the given solution of the initial value problem for t>0.

Concept explainers

Videos

To find: The solution of the initial value problem for t>0 and plot the graph for the given solution.

To Sketch: The graph for the solution curve P(t) when k=0.025, A=300 and P0=1 .

To describe: The long term behavior of the solution.

Chapter 9 Solutions

T o find: The solution of the initial value problem for t > 0 and plot the graph for the given solution.