Visualizing Differential Equations

Visual exploration can add significant meaning and promote a more general and intuitive understanding of many topics in differential equations.  When students can visualize the concepts they are learning, they will understand these concepts more deeply, and find it easier to remember them and connect them to related topics.  In differential equations, we have a number of opportunities to help our students visualize the topics we teach.

This page is designed to highlight some of the available resources on this site that can be used in teaching and learning Differential Equations.

Exploration Apps

• Direction Field Explorer
  This app will plot the direction field for any first-order differential equation that can be solved for $y’$.  A general solution can then be entered with a parameter $C$ and this parameter can then be varied to show how the solution curves will always follow the flow of the direction field vectors (or slope segments of the slope field, if this option is specified). The variables can be adjusted to match the application being studied for many different common variable names.  Clicking on the direction field will display a numerically generated solution curve through the selected point.
• CalcPlot3D
  This app allows the exploration of the phase portraits and solutions of systems of differential equations, both linear and non-llinear, and both autonomous and non-autonomous.  That is, you can even include a time parameter $t$ in the equations in your system.  Using parameter animation, CalcPlot3D will even show how the corresponding direction field changes as time passes and will display a corresponding solution curve when the user clicks on the plot.
  
  Since we spend more time on autonomous systems, let’s discuss what else can be done with them.  In addition to plotting the phase portrait, the user can Add a Space Curve to the phase portrait plot to represent the parametric form of the general solution of the system of differential equations.  In CalcPlot3D we can select from the parameters $a, b, c,$ and $d$ in CalcPlot3D to represent the parameters in the general solution.  Once this space curve is plotted on the phase portrait, the included parameters can be varied with sliders (or textboxes) to confirm that the solution curves always follow the vectors in the phase portrait.

Supplementary Textbook Content

• A Visual Introduction to Bifurcation, Section 2.4 Bifurcation in Prof. Seeburger’s custom OER Differential Equations textbook on the LibreTexts platform
• Video presentation by Prof. Seeburger about customizing an OER textbook for Differential Equations on the LibreTexts platform.

GeoGebra Simulations and Standalone Interactive Figures

• Spring Motion Simulation

  • Spring Motion Simulation (with/without Forcing)

• Bifurcation Motivation

  • Example 1, \(\dfrac{dy}{dt} = y^2 + 2y + a\),  Dynamic graph of $f(y) = y’$ to illustrate Bifurcation

• Bifurcation Diagrams

  • Example 1, $\dfrac{dy}{dt} = y^2 + 2y + a$,  Bifurcation Diagram
  • Bifurcation diagram for: $\dfrac{dy}{dt} =\frac{1}{50}(y^2 + ay + 16)$

• Laplace Transforms

  • Interactive Piecewise Function Plotter with the Unit Step Function

• Phase Portraits for Systems of Differential Equations

  • Example 1, Saddle Point
  • DE System Explorer, allows you to see the phase portrait of any 2 x 2 linear system of ODEs. 

See the GeoGebra Apps page for more Interactive Figures designed to help visualize Calculus and Differential Equations.