Of course, the slope field and solution graphs are standard topics in most differential equations courses. Since we are dealing with autonomous equations, we can capture all of the information provided by the slope field in a simpler, one-dimensional picture. In the phase line, circles indicate rest points, while arrows indicate the directions that solutions move as t increases. The existence and uniqueness theorem guarantees that solution curves do not cross the equilibrium points. Consequently, a phase line as depicted in Figure 2 implies that solutions between the equilibrium points p and q tend from q to p as t runs from plus to minus infinity.
Solutions tend from q to p as time increases Two quick comments are in order.
Also, these qualitative methods cannot tell us when a solution tends to infinity in finite time, but they do allow us to conclude that the solution is unbounded. One important reason for introducing the phase line early is the fact that much of our emphasis later in the course is on systems of equations rather than higher order equations. The phase line gives the students early practice doing this which serves them well in this later activity.