Problem 49 - Thomas Calculus

Question

Social diffusion Sociologists sometimes use the phrase "social diffusion" to describe the way information spreads through a population. The information might be a rumor, a cultural fad, or news about a technical innovation. In a sufficiently large population, the number of people \(x\) who have the information is treated as a differentiable function of time \(t,\) and the rate of diffusion, \(d x / d t\) , is assumed to be proportional to the number of people who have the information times the number of people who do not. This leads to the equation $$\frac{d x}{d t}=k x(N-x)$$ where \(N\) is the number of people in the population. Suppose \(t\) is in days, \(k=1 / 250,\) and two people start a rumor at time \(t=0\) in a population of \(N=1000\) people. a. Find \(x\) as a function of \(t .\) b. When will half the population have heard the rumor? (This is when the rumor will be spreading the fastest.)

Step-by-Step Solution

Answer

a) Solve the differential equation: \(x(t) = \text{function of } t\). b) When \(x=500\), solve for \(t\).

Step 1: Understand the Differential Equation

The differential equation given is \(\frac{dx}{dt} = kx(N-x)\). Here, \(x\) is the number of people who have received the information at time \(t\), \(N\) is the total population, and \(k\) is a constant. We know \(k = \frac{1}{250}\) and \(N = 1000\). We need to solve this differential equation for \(x\).

Step 2: Set the Initial Conditions

We are also given initial conditions: at time \(t = 0\), there are two people who know the rumor, so \(x(0) = 2\). The equation to solve becomes \(\frac{dx}{dt} = \frac{1}{250}x(1000-x)\).

Step 3: Solve the Differential Equation

The differential equation can be solved using separation of variables. We rearrange terms to get \(\frac{1}{x(1000-x)}dx = \frac{1}{250}dt\). Integrate both sides. The left side requires partial fraction decomposition: \(\frac{1}{x(1000-x)} = \frac{1}{1000} \left( \frac{1}{x} + \frac{1}{1000-x} \right)\), which can then be integrated.

Step 4: Perform Integration

Integrate the left side to get \(\frac{1}{1000} \left( \ln|x| - \ln|1000-x| \right)\). The right side integrates to \(\frac{t}{250} + C\). By applying properties of logarithms, you can simplify: \(\ln \left( \frac{x}{1000-x} \right) = \frac{1000}{250}t + C\).

Step 5: Apply Initial Conditions to Find C

Substitute the initial condition \(x = 2\) into \(\ln\left(\frac{2}{1000-2}\right) = \frac{1000}{250} \cdot 0 + C\) to find \(C\). Calculate \(\ln\left(\frac{2}{998}\right)\) to find \(C\).

Step 6: Express x in Terms of t

Using the equation \(\ln\left(\frac{x}{1000-x}\right) = \frac{4t}{1000} + C\), solve for \(x\) by exponentiating both sides: \(\frac{x}{1000-x} = e^{\frac{4t}{1000} + C}\). Solve this equation for \(x\) to express it as a function of \(t\).

Step 7: Calculate t When Half the Population Knows

For half the population, \(x = 500\). Substitute \(x = 500\) into the equation \(\frac{500}{500} = e^{\frac{4t}{1000} + C}\). Solve for \(t\), noting that \(e^{\frac{4t}{1000} + C} = 1\) when \(t\) gives half the population, leading to \(t = \left( \ln 1 - C \right) \frac{250}{4}\).

Key Concepts

Social Diffusion

Social diffusion is a term used to describe how information or ideas spread throughout a population. Imagine a rumor in a large group of people. Initially, only a few people know it, but as they share the information with others, the number who know grows. This spread depends on the number of people already aware compared to those still in the dark.
A differential equation can represent this situation. The rate at which people learn the information (\(\frac{dx}{dt}\)) is proportional to those in the know (\(x\)) and those who are uninformed (\(N-x\)) where \(N\) is the total population. This creates the equation \(\frac{dx}{dt} = kx(N-x)\). Understanding this equation allows us to predict how quickly a rumor will spread among a given population. In the exercise, the constant \(k\) represents the likelihood of spread/day, which affects how rapidly information diffuses.

Separation of Variables

The differential equation you see in social diffusion problems is solved using a technique called "Separation of Variables." This method involves rearranging the equation so that all terms involving one variable are on one side, and all terms involving the other are on the opposite side. This separation allows us to integrate each side independently to find a solution.
For example, take the equation \(\frac{dx}{dt} = \frac{1}{250}x(1000-x)\). By separating variables, it becomes \(\frac{1}{x(1000-x)}dx = \frac{1}{250}dt\). Each side is then ready for integration. This step is crucial as it transforms the differential equation into a form that can be further processed using calculus techniques to find a specific function that describes how information spreads over time.

Initial Conditions

Initial conditions are essential for finding specific solutions to differential equations. They serve as the starting point for the evolution of the system over time. In a social diffusion context, an initial condition might be the number of people who initially know the rumor. This detail is critical because it influences the integration constant when solving the equation.
In the exercise, at \(t=0\) (the beginning of our observation), two people already know the rumor. Thus, our initial condition is \(x(0) = 2\). This condition is used to find the constant of integration (denoted as \(C\)) after integrating the separated variables. Applying the initial conditions ensures the solution fits the specific scenario under analysis.

Logarithmic Integration

Logarithmic integration comes into play when integrating expressions that involve terms such as \(\frac{1}{x}\). This type of integration is pivotal in solving differential equations like the one in the social diffusion model.
For instance, consider integrating \(\frac{1}{x(1000-x)}\). Using partial fraction decomposition turns this into a sum of simpler fractions, \(\frac{1}{1000}\left(\frac{1}{x} + \frac{1}{1000-x}\right)\). Each of these terms integrates to a logarithmic function. The result of integrating provides a natural log expression, \(\ln\left(\frac{x}{1000-x}\right)\). This step is essential for transforming the separated differential equation into a functional form, which describes how the rumor spreads within the population over time.