Question 7.3.2: Using a Direction Field to Visualize the Behavior of Solutio...
Using a Direction Field to Visualize the Behavior of Solutions
Construct the direction field for
y^{\prime}=x+e^{-y}.
There’s really no trick to this; just draw a number of line segments with the correct slope. Again, we let our CAS do this for us and obtained the direction field in Figure 7.12a (on the following page). Unlike example 3.1, you do not know how to solve this differential equation exactly. Even so, you should be able to clearly see from the direction field how solutions behave. For example, solutions that start out in the second quadrant initially decrease very rapidly, may dip into the third quadrant and then get pulled into the first quadrant and increase quite rapidly toward infinity. This is quite a bit of information to have determined using little more than elementary algebra. In Figure 7.12b, we have plotted the solution of the differential equation that also satisfies the initial condition y(−4) = 2. We’ll see how to generate such an approximate solution later in this section. Note how well this corresponds with what you get by connecting a number of the line segments in Figure 7.12a.
