The Initial Value Problem y' = j(x, y); y(c) = d 39
I EXAMPLE 3 Direction Field for y' = .jXY
Graph the direction field of the differential equation y' = vxy = f(x, y) on
the rectangle R = { ( x, y) I - 5 :::; x :::; 5 and - 5 :::; y :::; 5}.
SOLUTION
To graph this direction field, we entered f(x, y) = .jXfj and set Xmin =
-5 , Xmax = 5, Ymin = -5 , and Ymax = 5. The graph of this direction
field and some of its solution curves are shown in Figure 2.4. Observe thatthe function f(x, y) = .jXfj is undefined in the second and fourth quadrants
where xy < 0. Since the differential equation y' = .jXfj is undefined for
xy < 0, there can be no solution to an initial value problem consisting of this
differential equation and an initial condition which corresponds to a point inthe second or fourth quadrant. Notice that the curve y = 0 (the x-axis) is a
solution of the differential equation. In the first and third quadrants xy > 0 ,
so y' = vxy > 0 ; and, therefore, all solutions in the first or third quadrant
are increasing functions. A solution which is in the third quadrant increases
until it terminates at the y-axis, or it increases until it reaches the x-axis.
If a solution in the third quadrant reaches the x-axis, it may continue along
the x -axis indefinitely or, once x becomes positive, it may increase into and
through the first quadrant. This is an example of a differential equation with
an infinite number of solutions passing through some points. Any point in the
third quadrant above the curve y = x^3 /9 and any point in the first quadrant
below the curve y = x^3 /9 has an infinite number of solutions passing through
it.
42y(x) O-2-4-4 -2 0
x2 4Figure 2.4 Direction Field and Solution Curves for y' = .jXfj