AP Calculus (AB)
Slope Fields

A slope field for a first order differential equation

is a plot of short line segments with slopes f(x, y) for a lattice of points (x, y) in the plane. [Recall that a differential equation is simply an equation that contains a derivative.]

Example 1:
Given , find the family of functions that solve this differential equation.
Solution:
Solve the differential equation by the "separation of variables" technique.

Thus, the solution is the set of all parabolas of this form, shifted up or down along the y-axis, depending on the value of C. Several of these parabolas are graphed below:

The slope of lines tangent to the original function will not be affected by the value of C. These tangent line slopes come from the equation y' = 2x – 1. Constructing a table of values for the derivative is a simple procedure:

If we now went to a graphing grid, we could sketch in as short segments these slopes and create a "slope field." This would indicate the direction of the graph of the original function (thus, a slope field is also called a "direction field"). The free software WinPlot has a feature that allows us to create these fields nearly instantly.

Further, if we are given a point that the graph passes through (an initial condition), we can find a specific solution to the differeential equation.

Given , and y(2) = 0, find the original function y.

and since y(2) = 0, then solving for C yields the following ...

Turning to WinPlot again, we can add the point the graph is known to pass through along with the specific solution.

Example 2:

Solve , with the initial condition y(2) = 1.