Graphical Solution Using Slope Fields

Many differential equations are impossible to solve by obtaining an explicit formula for the solution. Despite this impediment, much can be learned about the solution through a graphical approach using slope fields, which are sometimes referred to as direction fields.

Sketching Slope Fields of General and Particular Curves

image of graphed slope fields A slope field is a graph of short tangent line segments centered at each of many grid points (x, y) that approximates a family of solution curves to y' = f(x, y). Any solution curve must pass through points (x, y) and must have slope f(x, y) at each point. Drawing short tangent line segments with slope f(x, y) at multiple points (x, y) enables us to visualize the general shape and overall behavior of these solution curves. Note that solution curves are usually drawn in both directions from a starting point, corresponding to increasing and decreasing values of x. image of 4 graphed sloped fields When the slope field of a differential equation y'= f(x, y) is accompanied by an initial condition y(x₀) = y₀, the initial-value problem is readily solved since y(x₀) = y₀ indicates where in the slope field to begin drawing - at the point (x₀, y₀). The resulting curve represents the solution to the initial-value problem.

View the presentation below introducing how to draw a slope field for a given derivative and general and particular solutions of a differential equation. Depending on the given differential equation, drawing a slope field can be a tedious process; however, it is a required skill for the AP Calculus Exam.

Explore more of how slope fields are created for various differential equations by CLICKING HERE.

The slope field calculator for first order differential equations may be helpful in deepening your conceptual and procedural understanding. To access a slope field calculator CLICK HERE.

Several options exist for using technological tools to draw slope fields: Winplot software, a direction fields (slope fields) program for the TI-83/84Plus family, and the built-in functionality of TI-Nspire CX. Although Winplot software is a general purpose plotting utility for Windows. Recall from the Introduction to AP Calculus AB module that free connectivity software (TI-Connect) and low cost connectivity cables for the TI-83Plus/84Plus series are available for capturing screen shots of your calculator window and transferring programs from computer to calculator.

The next presentation illustrates how to use the TI-Nspire CX handheld device to graph the slope field of a particular solution curve through a given point and uses this to solve a real-life wildlife population problem. Begin viewing the video at 0:42.

Graphical Solution Using Slope Fields

Graphical Solution Using Slope Fields: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of graphical solutions using slope fields of general and particular solutions of differential equations.

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