a Explanation Given: The number of people who are infected with the diseases is model by the logistic equation d P d t = k P ( 1 − P A ) , P ( 0 ) = P 0 . Where k is a positive infection rate, A is the number of people in the community, and P 0 is the number of inflected people at t = 0 . Definition used: If the differential equation can be written in the form g ( y ) y ′ ( t ) = h ( t ) in which the terms that involve y appear on one side of the equation separated from the terms that involve t . An equation that can be written in this form is said to separable. Calculation: Consider the logistic equation d P d t = k P ( 1 − P A ) implies 1 P ( 1 − P A ) P ′ ( t ) = k . Therefore, the above differential equation is separable. On further simplification, the above equation becomes, 1 P ( 1 − P A ) d P d t = k d P P ( 1 − P A ) = k d t A P ( A − P ) d P = k d t ( 1 P + 1 A − P ) d P = k d t Integrate both the sides. ∫ ( 1 P + 1 A − P ) d P = ∫ k d t ∫ d P P + ∫ d P A − P = k ∫ d t ln | P | − ln | A − P | − ln C = k t + ln C , where C is the arbitrary constant . ln | P | − ln | A − P | − ln C = k t On further simplification, the above equation becomes, ln | P C ( A − P ) | = k t + C P C ( A − P ) = e k t P ( A − P ) = C e k t Substitute t = 0 b. To determine To sketch: The graph of the solution for k = 0.025 , A = 300 and P 0 = 1 . c. To determine To describe: The long-term behavior of the solutions for any P 0 with 0 < P 0 < A .

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 7.9, Problem 42E

To determine

To find: The solution of initial value problem in terms of k, A and P0.

To determine

To sketch: The graph of the solution for k=0.025, A=300 and P0=1.

To determine

To describe: The long-term behavior of the solutions for any P0 with 0<P0<A.

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Chapter 7.9, Problem 41E

Chapter 7.9, Problem 43E

Chapter 7 Solutions

Calculus: Early Transcendentals (2nd Edition)

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The solution of initial value problem in terms of k, A and P 0 .

Solution Summary: The author explains that the logistic equation is separable if the terms that involve k, A, and t are separated from each other.

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To find: The solution of initial value problem in terms of k, A and P0.

To sketch: The graph of the solution for k=0.025, A=300 and P0=1.

To describe: The long-term behavior of the solutions for any P0 with 0<P0<A.

Want to see the full answer?

Chapter 7 Solutions