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 9.2.24

Chapter 9, Problem 9.2.24

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

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Identify the given differential equation: \(P\'(t) = 0.05P - 0.001P^{2}\). This is a logistic differential equation modeling population growth with a carrying capacity.

Find the equilibrium solutions by setting the derivative equal to zero: \(0 = 0.05P - 0.001P^{2}\). Solve for \(P\) to find the constant solutions where the population does not change.

Rewrite the equation in the standard logistic form: \(P\'(t) = rP\left(1 - \frac{P}{K}\right)\), where \(r\) is the growth rate and \(K\) is the carrying capacity. Identify \(r\) and \(K\) from the given equation.

Sketch the direction field by analyzing the sign of \(P\'(t)\) for values of \(P\) less than, equal to, and greater than the equilibrium points. This will show whether the population is increasing or decreasing in those regions.

For each initial condition \(P(0) = 10\), \(P(0) = 40\), and \(P(0) = 80\), draw the qualitative solution curves on the direction field, showing how the population evolves over time toward the equilibrium solutions.

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Key Concepts

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

Logistic Differential Equation

A logistic differential equation models population growth with a carrying capacity, combining exponential growth and a limiting factor. It has the form P' = rP(1 - P/K), where r is the growth rate and K is the carrying capacity. This equation captures how growth slows as the population approaches K.

Equilibrium Solutions

Equilibrium solutions occur where the derivative of the function is zero, meaning the population does not change over time. For logistic equations, these are found by setting P' = 0 and solving for P. Equilibria indicate stable or unstable population levels depending on the system dynamics.

Direction Fields and Solution Curves

Direction fields graphically represent the slope of solutions at various points, helping visualize the behavior of differential equations. Solution curves are drawn following these slopes from given initial conditions, showing how the population evolves over time. They illustrate stability and long-term trends.

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Related Practice

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

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

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

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

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.

y'(t) = 2t²/(y² − 1), y(0) = 0

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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