Two trajectories approach equilibrium Show that the two

Chapter 9, Problem 7E

(choose chapter or problem)

Problem 7E

Two trajectories approach equilibrium Show that the two trajectories leading to ( m/n, a/b ) shown in Figure 9.31 are unique by carrying out the following steps.

a. From system (1a) and (1b) apply the Chain Rule to derive the following equation:

b. Separate the variables, integrate, and exponentiate to obtain ,

where K is a constant of integration.

c. Let also shown in Figure 9.35.

d. Consider what happens as (x, y) approaches( m/n, a/b).Take limits in part (b) as  to show that either

or  Thus any solution trajectory that approaches ( m/n, a/b ) must satisfy

e. Show that only one trajectory can approach ( m/n, a/b )from below the line  From Figure 9.35 you can see that ƒ(y0) < My, which implies that

This in turn implies that

Figure 9.35 tells you that for there is a unique value x0 < m/n satisfying this last inequality. That is, for each y <a/b there is a unique value of x satisfying the equation in part (d). Thus there can exist only one trajectory solution approaching ( m/n, a/b ) from below, as shown in Figure 9.36.

f. Use a similar argument to show that the solution trajectory leading to( m/n, a/b ) is unique ify0> a/b.