1550078481-Ordinary_Differential_Equations__Roberts_

532 Ordinary Differential Equations

.. PORTRAIT 1!!1001£3

Enter lh'n~yttem definin'g functions Fl. F2 Enter the number of initial value problems to :solve

::: : :: : ;:(~=·~:=·~==~~=~:,=.y2=========~

N • 2

For antl<the each initial initial conditions value problem dl i and i. enter d2i. the interval'of integration rai. bi]. the initial value ci.

;.. ai• bi.. ci• dli"' d2i ...

Figure B.6 Phase-Plane Graph of Y2 versus Y1 for

System IVPs (3)(i) and (3)(ii)

The following example illustrates how to use the RERUN current IVP op-

tion of PORTRAIT to generate a visually informative phase-plane graph.

EXAMPLE Producing a Phase-Plane Portrait

Use PORTRAIT to produce a phase-pla ne portrait of the system

(4)

SOLUTION

dx 2

dt = 3y-2x - 1

dy

dt

8x -y^2 - 7

To find the critical points of system ( 4), we set the right-hand side of both

components of (4) equal to zero. That is , to find the critical points of (4), we

simultaneously solve the following system

3y-2x^2 - 1 = 0

(5)

8x -y^2 - 7 = 0

Solving the first equation in (5) for y , we find

2 2 1

(6) y = -x + -.

3 3