The Initial Value Problem y' = f(x, y) ; y(c) = d 35(3) If the direction lines immediately above and to the left of (x, y) have
negative slope (y' < 0 above and to the left of (x, y)), then the solution
through the point (x , y) decreases as it approaches (x, y) from the left.
( 4) If below and to the right of (x, y) the direction lines have negative slope,
then the solution through (x, y) decreases to the right of (x, y).
(5) If above and to the right of (x, y) the direction lines have positive slope,
then the solution through (x, y) increases to the right of (x, y).• When f ( x, y) = 0 and ( 2) and ( 4) both occur, then there is a relative
maximum at, or near, (x, y).• When f(x, y) = 0 and (3) and (5) both occur, then there is a relative
minimum at, or near, (x , y).• When f(x, y) = 0 and either (2) and (5) both occur or (3) and ( 4) both
occur, then there is an inflection point at, or near, (x, y).
In order to use a computer program to graph the direction field of y' =f(x, y) in a rectangular region R of the xy-plane bounded by the lines x =
Xmin, x = Xmax, y = Ymin, and y = Ymax, you must enter the function
f and the values for Xmin, Xmax, Ymin, and Ymax. The way in which
you do this depends upon the software you are using. The following example
illustrates the typical output of such computer programs.I EXAMPLE 1 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 and - 5 :::; y :::; 5}.
SOLUTION
We input the function f(x, y)- namely, x - y and indicated Xmin = -5,
Xmax = 5, Y min = -5, and Y max = 5. The direction field shown in Figure 2 .1
is the output from MAPLE.
Notice that on the line y = x, y' = x -y = 0. Above the line y = x, y > x
so y' = x -y is negative. Thus, any solution which passes through some point
above the line y = x decreases until it reaches a minimum, which occurs when
the solution crosses the line y = x (where y' = 0). Once the solution crosses
the line y = x, y' = x - y becomes positive, since x > y and the solution
increases.
Look carefully at the direction field in Figure 2.1. You should be able topick out the line y = x - 1. That is, from Figure 2.1 it looks like y = x - 1
may be a solution of the differential equation y' = x - y. To determine