Question: The concentrations of two chemicals A and B as functions of time are denoted by x and y respectively. Each alone decays at a rate proportional to its concentration. Put together, they also interact to form a third substance, at a rate proportional to the product of their concentrations. All this is expressed in the equations:
(dx/dt) = -2x - xy, (dy/dt) = -3y - xy
(a) Find a differential equation describing the relationship between x and y, and solve it.
(b) Show that the only equilibrium state is x = y = 0. (Note that the concentrations are nonnegative.)
(c) Show that when x and y are positive and very small, y2/x3 is roughly constant. [Hint: When x is small, x is negligible compared to ln x.] If now the initial concentrations are x(0) = 4, y(0) = 8:
(d) Find the equation of the phase trajectory.
(e) What would be the concentrations of each substance if they become equal? (f) If x = e-10, find an approximate value for y.