I have always assumed that slope fields (direction or gradient fields) were relatively easy, and still do. Yesterday while working with a colleague I came across what could be a confusing point. Consider the function y=e^x. This function satifies the differential equation y'=y (1) and also satisfies the differential equation y'=e^x (2). However, if you sketch the slope field for each of these two diff eqs you get very different slope fields. The first, since y' is a function of y alone, has constant slopes along horizontal lines. The second, since y' is a function of x alone, has constant slopes along vertical lines.
The family of curves that satisfiy (1) is y=Ae^x, where A is an arbitrary constant. The family that satisfiy (2) is y=e^x+C where C is arbitrary. It turns out that y=e^x is in both families.
The confusion arose when a fellow tired to reverse the usual problem: Given a function (in this case y=e^x) find the slope field that this function "lives in". The answer is not unique.
This might not be a major point, but it is the first time I've seen some confusion with this.
Doug Kuhlmann Math Department Chair Phillips Academy 180 Main Street Andover, MA 01810 firstname.lastname@example.org