{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

1. 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$.
2. 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: \(\frac{dy}{y(1 - y)} = k \, dt\).
3. 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 \(\int \left( \frac{A}{y} + \frac{B}{1 - y} \right) dy = \int k \, dt\).
4. 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.
5. 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.

Key Concepts

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.