1550078481-Ordinary_Differential_Equations__Roberts_

422 Ordinary Differential Equations

V1 = G) and v2 = ( ~).

The eigenvectors of this example are identical to the eigenvectors of example 1,
but the eigenvalues have opposite sign. So the phase-plane portrait for this
system looks exactly like the phase-plane portrait for example 1 (Figure 10.4)
except that the arrows on the trajectories, which indicate the direction that a
pa rticle will take, must be reversed. In this instance, the origin is an unstable
critical point and as t , oo, the components x(t) , ±oo and y(t) _, ±oo.
Figure 10.4 with the direction arrows reversed is a typical phase-plane portrait
for an unstable node.

EXAMPLE 3 Determining the Type of Stability and

Sketching a Phase-Plane Portrait

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

portrait of the system

(8)

SOLUTION

x' = x + y

y' = 4x - 2y.

The eigenvalues and associated eigenvectors of the matrix

A=(! -~)

are >-1 = -3, >-2 = 2,

Hence, the general solution of (8) is

The vector v 1 is a vector with its "tail" at the origin and its "head" at (1, -4).

The vector v 1 and the line m 1 containing v 1 is drawn in Figure 10 .5. The
vector v 2 and the line m 2 containing v 2 is also drawn in Figure 10.5.