Two chemicals A and B react to form the substance X according to the law dx /dt = k(100 - x) (50- x), where k is a positive constant and x is the amount of substance X. Describe the amount of substance X, given the initial conditions in (a), (b), and (c). Sketch the direction field, equilibrium solutions, phase line, and solution starting from the initial value. Discuss the relative merits of each choice of initial conditions. Might a DE student question the validity of this model? Why?
(a) x(0) = 0
(b) x(0) = 75
(c) x(0) = 150