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Peer Review

Phase Portraits for ODE's and Autonomous Systems

by Richard Mansfield


Overall Numeric Rating:

4.5 stars
Content Quality: 5 stars
Effectiveness: 4 stars
Ease of Use: 4.25 starsstar
Reviewed: Jun 29, 2008 by Mathematics
Overview: The applet is designed to plot direction fields and solution curves for ordinary differential equations and autonomous systems of ODEs.
Type of Material: Simulation.
Recommended Uses: In-class presentation tool.
Technical Requirements: A Java-enabled browser.
Identify Major Learning Goals: The learning goal of this site is to illustrate the concept of a direction field and its associated flow lines as used in the theory of ordinary differential equations.
Target Student Population: Ordinary Differential Equations students.
Prerequisite Knowledge or Skills: Basic knowledge of ODEs.

Evaluation and Observation

Content Quality

Rating: 5 stars
Strengths: The applet is simple but informative. The simple interface allows the user to easily input two functions of x, y, and t to form a two dimensional system of differential equations (or select an example from a drop-down menu), then the corresponding direction field is sketched. The user can then select an initial point, using the mouse, and the flow line through that point is sketched. A separate solutions window plots the two solution curves (in different colors) corresponding to the last click in the phase portrait window. Two extra buttons on the interface allow the user to continue the solution curve for larger values of the time variable. There are instructions on how to use the applet as well as explanations of the underlying mathematics. The graphical display is very clear and looks like a standard DE textbook illustration. Window settings for the graph can be entered independently.
Concerns: The site containing the applet would benefit from some introductory paragraphs (outside of the main help window) on direction fields and autonomous systems.

Potential Effectiveness as a Teaching Tool

Rating: 4 stars
Strengths: The applet does one thing, and does it well. As a support and demonstration tool, instructors can use the site to illustrate textbook examples and homework assignments. Both the phase portrait and the solutions windows will expand to fill the entire screen: a big plus for classroom demonstrations. Students could benefit from visiting the site and spending 20-30 minutes checking their homework or working through a planned worksheet.
Concerns: An independent reader may very well have some difficulty with the exercises if his/her only preparation is reading the text on the website. Both graphs could benefit from the labeling of axes. The solutions graph, in particular, needs an indicator of which color represents which function.

Ease of Use for Both Students and Faculty

Rating: 4.25 stars
Strengths: This is a very straightforward applet to use. An average user should be able to run it immediately. The expression syntax is very similar to the syntax used by most graphing calculators and common computer algebra systems.
Concerns: It is impossible to close separate windows without completely exiting the site. Also, the phase portrait window constantly falls into background when a user is changing the input which is somewhat inconvenient.