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 9.3.46a

Chapter 9, Problem 9.3.46a

{Use of Tech} Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0≤y≤1, knows the rumor, while the remaining fraction 1−y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y1−y. Therefore, the equation that describes the spread of the rumor is y′(t)=ky(1−y), for t≥0 where k is a positive real number and t is measured in weeks. The number of people who initially know the rumor is y(0)=y0, where 0≤y0≤1.
a. Solve this initial value problem and give the solution in terms of k and y0.

Verified step by step guidance

Recognize that the given differential equation is a logistic differential equation of the form \(y'(t) = k y (1 - y)\), where \(k > 0\) and \(0 \leq y \leq 1\). The initial condition is \(y(0) = y_0\).

Separate the variables to prepare for integration: write the equation as \(\frac{dy}{dt} = k y (1 - y)\), then rearrange to isolate \(y\) terms on one side and \(t\) terms on the other: \(\frac{dy}{y(1 - y)} = k \, dt\).

Use partial fraction decomposition to express \(\frac{1}{y(1 - y)}\) as \(\frac{A}{y} + \frac{B}{1 - y}\), then find constants \(A\) and \(B\). This allows you to rewrite the integral as \(\int \left( \frac{A}{y} + \frac{B}{1 - y} \right) dy = \int k \, dt\).

Integrate both sides: the left side will involve logarithmic functions of \(y\) and \(1 - y\), and the right side integrates to \(k t + C\), where \(C\) is the constant of integration.

Solve the resulting equation for \(y\) explicitly in terms of \(t\), \(k\), and the initial condition \(y_0\) by applying the initial condition \(y(0) = y_0\) to find \(C\), and then isolating \(y\) to get the logistic function solution.

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Key Concepts

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

Logistic Differential Equation

The logistic differential equation models growth with a limiting factor, expressed as y'(t) = ky(1 - y). It describes how the rate of change depends on both the current amount y and the remaining capacity (1 - y), leading to an S-shaped growth curve. This is essential for modeling phenomena like rumor spread where saturation occurs.

Separation of Variables

Separation of variables is a method to solve differential equations by rewriting them so that each variable and its differential are on opposite sides. For the logistic equation, this allows integration of terms involving y and t separately, facilitating the derivation of an explicit solution for y(t).

Initial Value Problem (IVP)

An initial value problem specifies the value of the unknown function at a starting point, here y(0) = y0. Solving the IVP means finding a particular solution to the differential equation that satisfies this initial condition, ensuring the solution models the specific scenario accurately.

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Initial Value Problems

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Textbook Question

{Use of Tech} Analytical solution of the predator-prey equations The solution of the predator-prey equations

X'(t) = -ax + bxy,y’(t) = cy - dxy

can be viewed as parametric equations that describe the solution curves. Assume a, b, c, and d are positive constants and consider solutions in the first quadrant.

a. Recalling that dy/dx = y(t)/x′(t), divide the first equation by the second equation to obtain a separable differential equation in terms of x and y.

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Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].

y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

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Textbook Question

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:

a. Assume t = 0 corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and 2015, the population is given by P(t) = P(0)exp(rt). Estimate the growth rate r using this assumption.

134

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Textbook Question

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.

a. Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁eᵗ + C₂e⁻ᵗ. You may assume this function is the general solution.

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Textbook Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

a. Write an initial value problem that models the mass of the drug in the blood, for t ≥ 0.

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.

A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

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