Question 7.3.1: Constructing a Direction Field Construct the direction field...
Constructing a Direction Field
Construct the direction field for
y^{\prime}=\frac{1}{2} y . (3.2)
All that needs to be done is to plot a number of points and then through each point (x, y), draw a short line segment with slope f (x, y). For example, at the point (0, 1), draw a short line segment with slope
y^{\prime}(0)=f(0,1)=\frac{1}{2}(1)=\frac{1}{2} .Draw corresponding segments at 25 to 30 points. While this is a bit tedious to do by hand, a good graphing utility can do this for you with minimal effort. See Figure 7.11a for the direction field for equation (3.2). Notice that equation (3.2) is separable. We leave it as an exercise to produce the general solution
y=A e^{\frac{1}{2} x} .We plot a number of the curves in this family of solutions in Figure 7.11b using the same graphing window we used for Figure 7.11a. Notice that if you connected some of the line segments in Figure 7.11a, you would obtain a close approximation to the exponential curves depicted in Figure 7.11b. It is significant to note that the direction field was constructed using only elementary algebra, without ever solving the differential equation. That is, by constructing the direction field, we obtain a reasonably good picture of how the solution curves behave. Such qualitative information about the solution gives us a graphical idea of how solutions behave, but no details, such as the value of a solution at a specific point. We’ll see later in this section that we can obtain
approximate values of the solution of an IVP numerically.
