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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.R.26d
Chapter 9, Problem 9.R.26d

Logistic growth The population of a rabbit community is governed by the initial value problem
P′(t) = 0.2 P (1 − P/1200), P(0) = 50
d. What is the population when the growth rate is a maximum?

Verified step by step guidance
1
Recognize that the given differential equation models logistic growth: \(P'(t) = 0.2 P \left(1 - \frac{P}{1200}\right)\) with initial condition \(P(0) = 50\).
Recall that the growth rate \(P'(t)\) reaches its maximum when the derivative of \(P'(t)\) with respect to \(P\) is zero, because \(P'(t)\) depends on \(P\) directly.
Express the growth rate function as \(f(P) = 0.2 P \left(1 - \frac{P}{1200}\right)\) and find its critical points by differentiating with respect to \(P\): compute \(f'(P)\).
Set \(f'(P) = 0\) and solve for \(P\) to find the population values where the growth rate could be maximum or minimum.
Determine which critical point corresponds to the maximum growth rate by analyzing the sign of \(f'(P)\) around the critical points or using the second derivative test.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as the population approaches a carrying capacity. It is represented by the differential equation P'(t) = rP(1 - P/K), where r is the growth rate and K is the carrying capacity. This model captures limited resources affecting growth.
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Exponential Growth & Decay

Growth Rate and Its Maximum

The growth rate P'(t) represents how fast the population changes over time. To find when this rate is maximum, we analyze P'(t) as a function of P and determine the population size that maximizes it, often by setting the derivative of P'(t) with respect to P to zero.
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Intro To Related Rates

Initial Value Problem and Solution Behavior

An initial value problem specifies the starting population P(0) and governs the population's evolution over time. Understanding the initial condition helps predict the population trajectory and ensures the solution to the differential equation is unique and applicable to the given scenario.
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Initial Value Problems
Related Practice
Textbook Question

A second-order equation Consider the equation


t² y′′(t) + 2ty′(t) − 12y(t) = 0


b. Assuming the general solution of the equation is


y(t) = C₁ tᵖ¹ + C₂ tᵖ²,


find the solution that satisfies the conditions


y(1) = 0, y′(1) = 7

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Textbook Question

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

e. Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?

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Textbook Question

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.

d. For what values of A are the corresponding solutions decreasing, for t≥0

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Textbook Question

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = (2t+1)(y²+1)

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Textbook Question

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(x) = 4x csc y, y(0) = π/2

Textbook Question

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

c. Find the solution that satisfies the condition y(1) = 0