1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 411

line Li divides the xy-plane into two half-planes. In one half-plane dx/dt > O,

so x(t) is increasing in this region and the particle will tend to move to the

right. In the other half-plane dx/dt < 0, so x(t) is decreasing in this region

and the particle will tend to move to the left.

y

(u, q) • (p, q)

x

(0, -r /A)

Figure 10.l Graph of Li: -Cx + Ay + r = 0 for A> 0, C > 0, and r > 0

Let (p, q) be any point below and to the right of the line Li and let (u, q)
be the corresponding point on Li as shown in Figure 10.1. Since (u, q) is on
Li, -Cu+ Aq + r = 0. At the point (p, q)

dx

• = -Gp+ Aq + r = -Gp+ Aq + r - 0
  dt
  =-Gp+ Aq+ r - (-Cu+ Aq + r) = -C(p-u).

Since p is to the right of u, p - u > 0 and since we have assumed C > 0,

dx/dt = - C(p - u) < 0 in the half-plane below and to the right of the line

Li. In this region the particle tends to move to the left toward the line Li.
In the half-plane above and to the left of Li, dx / dt > 0 and in this region the
particle tends to move to the right toward the line Li. So the first nation in
the arms race tries, at all times, to adjust its expenditures by moving them
horizontally toward the line Li. (The reader should note that if the point
( u, q) is on the line Li, then dx / dt = 0 but most likely dy / dt -:/:- 0 on Li. If
dy / dt -:/:- 0 at ( u , q), the graph of y versus x will cross the line Li at ( u, q).)

A similar argument shows that (i) L 2 divides the xy-plane into two half-

planes, (ii) in the half-plane below and to the right of L 2 , dy/dt > 0 so y(t)

is increasing in this region and the particle tends to move upward toward the

line L 2 , and (iii) in the half-plane above and to the left of L2, dy / dt < 0,