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 y(1 - y). Therefore, the equation that models the spread of the rumor is y'(t) = ky(1 - y), where k is a positive real number. The fraction 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.

b. Assume k = 0.3 weeks^-1 and graph the solution for y₀ = 0.1 and y₀ = 0.7.

c. Describe and interpret the long-term behavior of the rumor function for any 0 < y₀ < 1.