1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 443

(iii) Use PORTRAIT to solve system (15) and the following five initial

conditions on the interval [O, 2]:

(i) x(O) = 0, y(O) = 3; (ii) x(O) = 0, y(O) = 7; (iii) x(O) = 3, y(O) = O;

(iv) x(O) = 3, y(O) = 10; (v) x(O) = 7, y(O) = 10.

Graph the phase-pla ne graph of y versus x on the rectangle with

0 :::; x :::; 10 and 0 :::; y :::; 10.

d. (i) Find the critical point in the first quadrant of the system

dx

- = y^2 -4x+4

(16)

dt

dy

• = x^2 - 4y+4.
  dt

(ii) Write the associated linear system and calculate the eigenvalues.

What can you say about the stability of this linear system? What can

you conclude about the stability of the nonlinear system (16) at the

critical point?

(iii) Solve the system initial value problems co nsisting of system (16)

and the following initial conditions on the interval [O, 3]:

(i) x(O) = 0, y(O) = O; (ii) x(O) = 0, y(O) = 4; (iii) x(O) = 3, y(O) = O;

(iv) x(O) = 3, y(O) = 4.

Display y versus x on the rectangle with 0 :::; x :::; 5 and 0 :::; y :::; 5.

What do you infer about the stability of the nonlinear system (16) from

these results?

Additional Information Needed to Solve Exercises 6 and 7

Recall that a homogeneous system of n linear first-order differential equa-
tions with constant coefficients can be written in matrix-vector notation as

(17) y'=Ay

where y is a n n x 1 vector and A is an n x n constant matrix. The origin,

y = 0 , is a critical point of this system. We state the following theorem

regarding the stability of the critical point at the origin.

STABILITY THEOREM

(1) If the eigenvalues of A all have negative real part, then the origin is

a n asymptotically stable critical point of the linear system (17).

(2) If any eigenvalue of A has positive real part, then the origin is an

unstable critical point of the linear system (17).