Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.5.1

Chapter 9, Problem 9.5.1

Explain how the growth rate function determines the solution of a population model.

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Understand that in population models, the growth rate function describes how the population changes over time, often expressed as a function of the current population size, say \(r(P)\), where \(P\) is the population at time \(t\).

Recognize that the population model is typically formulated as a differential equation of the form \(\frac{dP}{dt} = r(P) \cdot P\), where \(\frac{dP}{dt}\) represents the rate of change of the population with respect to time.

Analyze how the form of the growth rate function \(r(P)\) influences the behavior of the solution: for example, if \(r(P)\) is constant and positive, the population grows exponentially; if \(r(P)\) decreases as \(P\) increases, it may model limited resources leading to logistic growth.

Solve the differential equation by separating variables or using an integrating factor, depending on the form of \(r(P)\), to find the explicit solution \(P(t)\) that describes the population at any time \(t\).

Interpret the solution \(P(t)\) in terms of the growth rate function to understand long-term behavior such as equilibrium points, carrying capacity, or unbounded growth, which are all determined by the characteristics of \(r(P)\).

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Key Concepts

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

Growth Rate Function

The growth rate function describes how the population changes over time, often expressed as a rate of change dependent on the current population size. It can be constant or vary with population, influencing whether the population grows, declines, or stabilizes.

Differential Equations in Population Models

Population models are typically formulated as differential equations where the growth rate function defines the derivative of the population with respect to time. Solving these equations provides the population size as a function of time, revealing the dynamics of growth.

Equilibrium Solutions and Stability

Equilibrium solutions occur when the growth rate is zero, indicating a stable or unstable population size. Analyzing these points helps predict long-term behavior of the population, such as whether it will settle at a steady state or experience unbounded growth or decline.

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Textbook Question

33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

z(x) = (z² + 4)/(x² + 16), z(4) = 2

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Textbook Question

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = y³sin t, y(0) = 1

Textbook Question

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0

Textbook Question

y'(t) = 2t²/(y² − 1), y(0) = 0

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Textbook Question

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P − 0.001P²; P(0) = 10, P(0) = 40, P(0) = 80

Textbook Question

12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

y'(t) = 4−y, y(0) = −1