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(a-x)(b-x) where k > 0 is a constant of proportionality and b > a > 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 -> infinity. (b) consider the case when a = b. use a phase portrait of the differential equation to predict the behavior of x(t) as t -> infinity when x(0) < a. when x(0) > a. (c) verify that an explicit solution of the de in the case when k=1 and a=b is x(t)=a-1/(t+c). find a solution that satisfies x(0) = a/2. then find a solution that satisfies x(0)=2a. graph these two solutions. does the behavior of the solutions as t-> infinity agree with your answers to part (b)?

(a) We're given that . We have four cases to consider: (1) ; (2) ; (3) ; and (4) .

In case (1), both and , so , which means is increasing and converging to as .

In case (2), and again, so as well, so is increasing again and converging to from below.

In case (3), and , so , which means is decreasing and converging to from above as .

Finally, in case (4), both and , so tha t, and so is increasing and diverging as .

So a simplified phase portrait for the solution might look something like



where the vertical axis represents .

(b) If , then , which has the phase portrait



so that when ; from below when ; and from below as in either case of or .

(c) With and , we have the separable ODE








With the initial value , we have



With , we would get



In both cases, we're assuming that , since we are given that from the start. In part (b), we found that as (regardless of whether is smaller or larger than , in fact), so we only need to verify that the solution we found also conforms to this trend. We have



which confirms the conclusion of the phase portrait.