The concentrations of two chemicals A and B as functions of time are denoted by x and y

Chapter 0, Problem 51

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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 = e10, find an approximate value for y.