1550078481-Ordinary_Differential_Equations__Roberts_

566 Ordinary Differential Equations

3. a. (3, 2), (4.22457, 2.9 83 00)

b. At (3, 2) the associated linear system

x' = -6(x - 3) + 9(y - 2)

y' = 4(x - 3) - 4(y - 2)

has an unstable saddle point. So (3, 2) is an unstable critical point of

the nonlinear system (11).

At (4.22457, 2.98300) the associated linear system

x' = -8.44914(x - 4.22457) + 9(y - 2.98300)

y' = 4( x - 4.22457) - 5.966(y - 2.98300)

has an asymptotically stable node. So ( 4.22457, 2.98300) is an asymp-

totically stable node of t he nonlinear system (11).

5. a. A parabola with vertex at (r/C, 0), axis of symmetry t he x-axis , and
   opens to the right. A parabola with vertex at (0, s/ D), axis of sym-
   metry the y-axis, and opens upward. Four. Two.

b. AB^2 x^4 + 2ABsx^2 - CD^2 x + As^2 + rD^2 = O

c. (i) (5, 4)

(ii) The associated linear system

x ' = -4(x - 5) + 8(y - 4)

y' = lO(x - 5) - 5(y - 4)

has an unstable saddle point at (5, 4). So (5, 4) is an unstable

critical point of t he nonlinear system (15).

d. (i) (2, 2)

(ii) The associated linear syst em is

x ' = -4(x - 2) + 4(y - 2)

y' = 4(x - 2) - 4(y - 2).

The eigenvalues of t he associated linear system are 0 and -8.

Nothing. Nothing.

(iii) The critical point is unstable.

7. a. (i) unstable b. (i) stable