Part 1: The Slope Field Concept
In this module we study a way to construct a graphical representation of a differential equation of the form
dP/dt = f(t, P).
Specific examples include
In principle, the slope function might depend on both t and P. However, the differential equations we have seen so far have right-hand sides that depend only on P, not on t. There is no harm in thinking of the slope function as depending on both t and P, even when only of these variables appears in the formula.
Recall from the Limited Population Growth module that the value of the slope function at any particular point in the (t, P)-plane is the slope at that point of a solution curve through that point. Here is the picture from that module that illustrated this
At (tk-1,Pk-1), we calculated an exact slope, and we treated that as the slope of a solution through that point and extending for a short run of Delta-t. That is, we treated the green line segment as a piece of the unknown solution curve through (tk-1,Pk-1). Now suppose we have a grid of closely spaced points covering some rectangle in the (t,P)-plane, as in the next figure:
At each point (t, P) of the grid we draw a "short green segment" (the color doesn't matter) whose slope is given by the expression f(t, P). The next figure shows the result of doing that for the specific differential equation
dP/dt = t - P.
Notice that this "field of slopes" -- or slope field -- shows very clearly the shapes of possible solution curves. A slope field is also called a direction field, since it shows the directions followed by solution curves. (Your helper application may use this name.)
Your answers to these questions demonstrate that you can get a lot of information about solutions directly from the slope field -- a picture of the problem -- without calculating any algebraic form for solutions. In fact, we even found an algebraic formula for one solution, and that one turned out to be important for describing all the others.
Last modified: October 5, 1997