1550078481-Ordinary_Differential_Equations__Roberts_

534 Ordinary Differential Equations

We chose the interval of integration to be small enough so that this trajec-
tory did not touch the x-axis- the lower edge of the rectangle R. Using this
technique, we can tell the direction of a trajectory which begins at one bound-
ary of R and also leaves at a boundary of R. When a trajectory touches the
edge of the rectangle, this is a point where the trajectory "begins." Whereas,
when the trajectory approaches the edge of the rectangle but does not touch
the edge, this indicates the direction the trajectory is headed with increasing
time. Any trajectory which starts on the positive y-axis below the value 2.5
will have a trajectory similar to the one shown in Figure B.7.

Next, we chose the point (0, 3) on the left edge of the rectangle R and
generated a numerical solution of the initial value problem consisting of the
system (4) and the initial condition (ii) x(O) = 0, y(O) = 3. We displayed
a phase-plane graph of this solution and saw that it proceeded toward the
critical point at (1, 1) and then abruptly changed direction toward the critical
point at (2, 3). Then we generated the solution to the IVP consisting of
system (4) and the initial condition (iii) x(O) = 0, y(O) = 5. This trajectory
begins in the upper left hand corner of the rectangle R and proceeds toward
the critical point at (2, 3). A graph of the trajectories of the initial value
problems (i) through (iii) are shown in Figure B.8. Any trajectory which
begins on the positive y-axis between the values of 2 and 5 has a trajectory
similar to the trajectories of the initial value problems (ii) and (iii).

'" PORTRAIT S~D

Enlet lhe syslem defining functions F1. F2

y'l •Fl • j3•yi.2' y1'y1-1

y'2. F2. J8'y1-y2'y2·7

Entm the number of initial value pJoblems·to solve

Horizontal Range

Hmin"' ,__ ___ ,

For and each the initii!I initial conditions value problem dl i and i, enter d2i. the interval of integration (aL bil the initial value ci. Hmax ..

Vmtical.Range

Vmm• I 0

Vmax•J 5

OPTIONS

i• a i .. bi• c i .. dli ..

1. 1 0
   0 '.'.5

d2i.

25

V~~m-~in~~-'-~~..;...~..,_m-..,_..,; ____ __., ____ ...:.. ____ ..,:_ ____ .,:.,... ____ :...., ___ H_,mu

Figure B.8 Phase-Plane Graph of System (4) Using

Initial Conditions (i), (ii), and (iii)