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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

05:03

Initial Value Problems

Related Practice

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.

views

Textbook Question

{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.

views

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.

views

Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.

y′(t) = 2y + 4

Textbook Question

46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t)+km(t)=I, where m(t) is the mass of the drug in the blood at time t≥0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I=10mg/hr and k=0.05 hr^−1.

a. Draw the direction field, for 0≤t≤100, 0≤y≤600.

views

Textbook Question

Consider the differential equation y'(t)+9y(t)=10.

a. How many arbitrary constants appear in the general solution of the differential equation?

views