Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled by the autonomous differential equation

dX/dt = k(α - X)(β - X),

where k . 0 is a constant of proportionality and β > α > 0. Here X(t) denotes the number of grams of the new compound formed in time t.

(a) Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞

(b) Consider the case when α = β. Use a phase portrait of the differential equation to predict the behavior of X(t) as t → ∞ when X(0) < α. When X(0) > α.

(c) Verify that an explicit solution of the DE in the case when k = 1 and α = β is X(t) = α – 1/(t + c). Find a solution that satisfi­es X(0) = α/2. Then fi­nd a solution that satisfi­es X(0) = 2α. Graph these two solutions. Does the behavior of the solutions as t → ∞ agree with your answers to part (b)?