1550078481-Ordinary_Differential_Equations__Roberts_

38 Ordinary Differential Equations

and the x and y axes are to appear as a box outside of the rectangle (axes =

BOXED ) instead of passing t hrough the origin perpendicular to one another.
You can change t he size and location of the rectangle by changing the range
of t he variables x and y. The fourth statement causes the graph shown in
F igure 2. 1 to be displayed on t he computer monitor.

I EXAMPLE 2 Direction Field for y' ~ -x/ y

Graph the direction field of the differential equation y' = -x/y = f(x, y)
on the rectangle R = { ( x, y) I - 5 :::; x :::; 5 an d - 5 :::; y :::; 5}.

SOLUTION

Notice that f(x, y) is undefined when y = 0- t hat is , f is undefined on t he

x-axis. So there is no solution which passes through (x , 0) for any x. To graph
the direction field on the rectangle R, we set f(x, y) = -x/y, Xmin = -5,
Xmax = 5, Ymin = -5, and Ymax = 5. The result ing graph is displayed in
F igure 2.3. From the graph we see that the solut ion curves above the x -axis
increase in the second quadrant, have a maximum at the y-axis , and decrease
in t he first quadrant. We also see from the graph that solution curves below
t he x-axis decrease in the third quadrant, have a minimum at the y-axis,
and increase in the fourth quadrant. The solution of the differential equation

y' = -x/y is discussed in detail in example 4 of the foll owing section.

4

2

y(x) 0

-2

-4

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-4 -2 0

x

2

F igure 2.3 Direction Field for y' = -x/y

4