1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 421

particle starts at some point not on line m 1 or line m 2 - that is, suppose

c1 =I 0 and c2 =I 0. Since x(t) = c 1 e-^3 t - c 2 e-^5 t ---+ 0 as t ---+ oo and since

y(t) = c 1 e-^3 t+2c2e-^6 t---+ 0 as t---+ oo, the particle proceeds toward the origin.

Since e-Gt approaches 0 faster than e-^3 t as t approaches oo, the trajectories

of particles not on lines m 1 and m2 approach (0, 0) asymptotic to the line

m 1 as shown in Figure 10.4. In this case, the origin is an asymptotically

stable critical point, since x(t) ---+ 0 and y(t) ---+ 0 as t---+ oo. The phase-plane

portrait of Figure 10.4 is typical of an asymptotically stable node.

4

2

y 0

-2

-4

-4 -2 0 2

x

Figure 10.4 Asymptotically Stable Node

EXAMPLE 2 Determining the Type of Stability and

Sketching a Phase-Plane Portrait

4

Determine the type of stability of the origin and sketch a phase-plane
portrait of the system

(7)

SOLUTION

x' = 4x - y

y' = -2x + 5y.

The eigenvalues and associated eigenvectors of the matrix

A=( 4 -1)

-2 5

are μ1 = 3, μ2 = 6,