### Slope Fields

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

• dP/dt = k P, which represents natural population growth (if k > 0) or radioactive decay (if k < 0).
• dP/dt = c P (M - P), which represents limited population growth.

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.)

1. What are the slopes along the line P = t in the figure above?
2. How do you calculate the slopes along the horizontal axis, P = 0?
3. How do you calculate the slopes along the vertical axis, t = 0?
4. What slopes do you find along the line P = t - 1? Explain why this function is a solution of the differential equation.
5. Is there any other linear function that is a solution? Explain.
6. Describe in words the solution functions (or the graphs of solution functions) other than the linear function in step 4. In particular, what happens to solution functions as t becomes large?
7. Your helper application file has been set up to draw a slope field like the one above, and to add three solution curves. Carry out the steps in the file, and make sure the results confirm your answers to the questions above.

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.

Send comments to the authors <modules at math.duke.edu>