Chemical Reactions When certain kinds of chemicals are combined, the rate at which a new

Chapter 2, Problem 40

(choose chapter or problem)

Chemical Reactions When certain kinds of chemicals are combined, the rate at which a new compound is formed is governed by the differential equation dX -= k(a - X)(/3 - X) dt ' where k > 0 is a constant of proportionality and f3 > a > 0. Here X(t) denotes the number of grams of the new compound formed in time t. See page 21. (a) Use a phase portrait of the differential equation to predict the behavior of X as t oo. (b) Consider the case when a = {3. Use a phase portrait of the differential equation to predict the behavior of X as t oo when X(O) < a. When X(O) > a. ( c) Verify that an explicit solution of the DE in the case when k = 1 and a= f3 is X(t) = a - ll(t + c). Find a solution satisfying X(O) = a/2. Find a solution satisfying X(O) = 2a. Graph these two solutions. Does the behavior of the solutions as t oo agree with your answers to part (b )?