AP Calculus
Definition of a
Differential Equation
Differential Equations and Slope Fields
A Differential equation is an equation involving the
variables x, y, and y’ (or other derivative of y).
The solution to a differential equation is called the GENERAL
SOLUTION. To find a PARTICULAR SOLUTION, there must be
some initial conditions given.
Example:
y’’ – y = 0 is a differential equation.
Is y = sin x a solution?
Example:
Is y = 4e-x a solution?
xy’ – 3y = 0 (differential equation) VERIFY that y=Cx3 is the
general solution.
Find the particular solution with the initial conditions of y=2 and
x=-3.
Introduction to Slope
Fields
Solving a differential equation analytically can be difficult (and
sometimes impossible). Slope Fields provides a GRAPHICAL
approach that provides information about the solution to a
differential equation.
A slope field is sometimes called a direction field. A slope field
can be constructed by drawing small segments that represent the
slope of the solution curve at several points. Given enough of these
segments, a visual perspective of the solution curve appears.
Example:
Draw a slope field for the differential equation y’ = x-y at the
points (-1,1), (0,1), and (1,1).
Examples of slope fields.
Use the following procedure in order to draw the slope field for a
Graphing Slope Fields on a given differential equation on the TI-89 Graphing Calculator:
TI-89
 Click on MODE – the cursor should be positioned on the
Graph menu item. (Note that “FUNCTION” is the default
setting).
 Use the right arrow key to display the other selections.
 Move the cursor so that “DIFF EQUATIONS” is highlighted
and then press ENTER twice.
 In the Y = screen, enter the expression for the differential
equation in Y1. Use “t” instead” of x in the expression and
use “Y1” instead of just Y. Hit ENTER.
 When you hit the GRAPH key, the slope field (sometimes
called the direction field) should be displayed. Change the
window as necessary for appropriate viewing/interpretation.
ZOOM Decimal might be a good starting place.
 If you wish to graph a specific solution based on an initial
condition, go back to the Y = screen and enter an initial
condition value on the line marked Yi1. When you hit enter,
the specific solution curve will be superimposed on the slope
field. [Keep in mind that the calculator will graph solution
curves as functions so that it will draw, for example, a
semicircle rather than a complete circle given a single initial
condition. ]