Slope fields are an important tool to investigate the behavior of first order differential equations. This applet allows the user to visualize the slope fields for two coupled first order differential equations in two unknowns. Solutions to the differential equation are shown wherever the user chooses to click the mouse as well as the isoclines of the differential equation. The applet is designed to teach students exactly what a slope field looks like and what orthogonal directories are. It could be used to see the behavior of y(x) of an unsolvable differential equation. Solutions to differential equations that model predator and prey problems, growth rate of bacteria problems, and other applications can be shown.
Using the Applet:
First enter the differential equation into the text field. Use asteriks to indicate multiplication signs and cos(x) and sin(x) to indicate cosines and sines.
Click the button plot and the program will automatically draw a slope field. Arrows indicate which direction the slope runs in and colors indicate magnitude (the darker the higher the magnitude of the slope .
In order to see the solution curve left click on the mouse on the point where you would like to see the solution curve drawn. and to see the orthogonal plots,
Right click on the mouse where you would like the orthogonal directory drawn.
To clear the graph press the reset button.
To change the parameters of the graph, merely type in the maximum parameter in the x and y direction and press plot. If you type in say 5 for xmax and 8 for ymax, then the graph will go from -5 to 5 in the x-direction and -8 to 8 in the y-direction
The Theory Behind Slope Fields
A first order differential equation dy/dt and dx/dt gives us, for any point (x,y) on the x, y plane, the slope of the solution curve. Just as a flag reveals the particular direction for the air current at a flagpole, dy/dt and dx/dt reveal the direction (slope) of a solution curve y=f(x) at any point (x,y) we choose. An aerial view of flags on a grid of flagpoles in a field would show the overall pattern of different wind currents in the field. Similarly, vectors with the slope dx/dt and dy/dt indicate the flow of solution curves. Although dxdt and dy/dt do not give us the solution y=f(x), it does find the slope of the solution curve at any point (West, 9) The aim of this program is to show that slope so that one can see the solution curve and see the solutions going through that curve. This program also graphs orthogonal directories. These are directories that are perpindicular to the slope. If dy/dx=f(x,y) then the orthogonal directories follow the solution to the equation dy/dx= -1/f(x,y). These directories are commonly used to represent equipotential lines, where the slope and the magnitude of the slope are constant. If one follows these directories the gradient of the curve is zero.(West, 9)
West, Beverly, Steven Strogatz, Jean Marie McDill, and John Cantwell. “Interactive Differential Equations.” Addison Wesley Interactive, 1996.