This applet allows the user to input a first order differential equation and to display its slope field as well as solution curves for given initial conditions. It is part of a series of Differential Equation labs that have been used successfully at Kansas State University for over 10 years.
The ability to use the slope field of a differential equation to predict the shape of the solution curves.
Target Student Population:
Students learning about first order differential equations.
Prerequisite Knowledge or Skills:
Students should understand derivatives and the definition of a differential equation.
Type of Material:
It could be used both in class for demonstrations and as a tool around which to structure homework assignments.
JAVA applet should run in any browser.
Evaluation and Observation
The applet displays slope fields, single integral curves for one user-specified initial condition, and/or an ensemble of integral curves for 31 pre-specified initial conditions. When it works (which it usually does), it is fast and accurate.
The applet can fail, and when it does it freezes the browser, requiring a page refresh to continue. For instance, given the equation y?=y^(1/3) the slope field is plotted only for y>=0, though it also exists for y<0. An attempt to plot a solution with y(-3)<0 freezes the browser, as does an attempt to plot the ensemble of solutions since some of them involve y(-3)<0. Also, for equations which legitimately do not possess slope fields for y in some range such as y?=sqrt(y^2-1), the ensemble plot freezes the browser as does an attempt to plot a solution with initial condition in the forbidden range.Also see ease of use.
Potential Effectiveness as a Teaching Tool
This lab is very versatile. Its successful use at KSU for over a decade impressively testifies to its quality.
One of the uses the author has in his book is an exercise in which students are to find an initial condition at x=-3 such that the solution is within a specific small range at x=4. This exercise forces students to use facts such as that solution curves will lay smoothly on the vectors of the direction field or such as that solution curves do not cross each other (under mild continuity conditions). In this fashion under the right guidance and with the right questions a ?trial and error? exercise makes students actively engage and use rather theoretical facts. An extension could be for students to devise a fastest possible automatic method to achieve the desired value at x=4. Invariably this would lead to bisection, thus showing that students should be ready to make connections between different parts of the curriculum.
Another possible use is the demonstration of stability of solutions. For an equation such as,
say y?=y(1-y) students could try to predict the shape of solution curves and one could ask them to try to find a curve that starts with y(-3)>0 and which does not converge to 1. The impossibility to find such a solution can lead to a discussion of how to predict from the DE that this will happen.
The author did not put any of the good ideas from his text into the web site. His book, which is currently used internally at KSU, can be obtained at low cost from the author.
Ease of Use for Both Students and Faculty
No-nonsense input of the function in an input box allows for use of the applet for just about any differential equation. Similarly, the slope field as well as solutions for specific initial conditions are available at the touch of a button.
The window is static [-3,4] by [-4,4]. The initial values have to be declared for y(-3). It would be nice to have more freedom to choose here. For those of us who have gotten used to other interfaces it takes a minute to remember the notation and the applet does not give any feedback on incorrect input (say, forgotten multiplication signs).
Documentation seems only available in the author?s book.