The Initial Value Problem y' = f(x, y); y(c) = d 37\ \
4
~ ~2 ~ ~
~ ~
\ \
y(x) 0-2-4~~~~ ~~ }.
}. }.
I IC=O -4 -2 c = -1 c = -2 (^0 2 4)
x
Figure 2.2 Direction Field and Solution Curves for y' = x - y
!Co=ents on Computer Software! The software which accompanies this
text contains a program named DIRFIELD. It graphs the direction field of the
differential equation y' = f(x, y) in the rectangular region R of the xy-plane
bounded by the lines x = Xmin, x = Xmax, y = Ymin, and y = Ymax.
To graph the direction field you need to enter an expression for the func-
tion f(x, y) and the values for Xmin, Xmax, Ymin, and Ymax. The output
of DIRFIELD is similar to the output shown in Figure 2.1. Complete in-
structions for using DIRFIELD are contained in Appendix A. Figure 2.1 was
produced using the following four MAPLE statements.
with(DEtools): with(plots):
de:=diff(y(x), x) = x - y(x):
p:=DEplot(de, y(x), x = -5 .. 5, y = -5 .. 5, arrows=LINE, axes=BOXED):
display(p);The first statement informs MAPLE what software packages are required
to run the program. The second statement specifies the differential equation
whose direction field is to be plotted- in this instance, ~~ = x -y. To specify
any other differential equation, the expression x - y( x) which appears in the
second statement must be replaced and written in the appropriate MAPLE
syntax. The third statement specifies that the x values for the graph are to
vary from -5 to 5, the y values for the graph are to vary from -5 to 5, the
direction field elements are to be graphed as line segments (arrows= LINE),