Chapter 9.2, Problem 48E

Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.

Chemical rate equations Consider the chemical rate equations y′(t) = −ky(t) and y′(t) = − ky²(t), where y(t) is the concentration of the compound for t ≥ 0, and k > 0 is a constant that determines the speed of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

1. a. Let k = 0.3 and make a sketch of the direction fields for both equations. What is the equilibrium solution in both cases?
2. b. According to the direction fields, which reaction approaches its equilibrium solution faster?