Mastering Differential Equations: The Visual Method

Overview

Embark on an amazing mathematical journey with this course comprised of 24 intellectually stimulating and visually engaging half-hour lectures. Differential equations represent the pinnacle of a mathematics education, typically requiring extensive memorization of formulas. However, using computers, we can approximate solutions and display them in easy-to-understand graphics. Professor Robert Devaney, a pioneer of the visual approach, guides you through this course, leveraging visual tools to help you master differential equations without the need for rote memorization. You’ll explore solutions visually and learn to calculate and display them even with simple tools like spreadsheets.

Course Instructor

Professor Robert Devaney, coauthor of a widely used textbook on ordinary differential equations, draws on his expertise to illustrate the geometric behavior of differential equations and their astonishing applications in various fields.

Video Lessons

• What Is a Differential Equation?
  • Duration: 32 min
  • Description: Understand the concept of differential equations as equations for missing mathematical functions.

2. A Limited-Growth Population Model
   • Duration: 30 min
   • Description: Investigate solutions to first-order differential equations using a logistic growth model.

• Classification of Equilibrium Points
  • Duration: 31 min
  • Description: Explore source, sink, and node equilibrium solutions that govern nearby solution behavior.

4. Bifurcations-Drastic Changes in Solutions
   • Duration: 30 min
   • Description: Discover how small parameter changes can lead to significant solution changes.

• Methods for Finding Explicit Solutions
  • Duration: 31 min
  • Description: Learn standard methods for solving linear and separable first-order equations.

6. How Computers Solve Differential Equations
   • Duration: 28 min
   • Description: Explore Euler’s method and its application for approximating solutions using computers.

7. Systems of Equations-A Predator-Prey System
   • Duration: 30 min
   • Description: Analyze a predator-prey relationship through systems of differential equations.

8. Second-Order Equations-The Mass-Spring System
   • Duration: 30 min
   • Description: Examine the dynamics of a mass-spring system using second-order differential equations.

9. Damped and Undamped Harmonic Oscillators
   • Duration: 33 min
   • Description: Understand Euler’s formula and its connection to harmonic oscillators.

10. Beating Modes and Resonance of Oscillators
    • Duration: 32 min
    • Description: Investigate periodic forcing in oscillators and its implications for real-world structures.

11. Linear Systems of Differential Equations
    • Duration: 30 min
    • Description: Review linear algebra tools for solving linear systems of differential equations.

• An Excursion into Linear Algebra
  • Duration: 33 min
  • Description: Learn about eigenvalues and eigenvectors in solving linear systems.

13. Visualizing Complex and Zero Eigenvalues
    • Duration: 32 min
    • Description: Understand complex eigenvalues and their implications using Euler’s formula.

• Summarizing All Possible Linear Solutions
  • Duration: 32 min
  • Description: Explore special cases of eigenvalues and visualize phase planes for linear systems.

15. Nonlinear Systems Viewed Globally-Nullclines
    • Duration: 32 min
    • Description: Start analyzing nonlinear systems using the nullcline method.

16. Nonlinear Systems near Equilibria-Linearization
    • Duration: 31 min
    • Description: Approximating behaviors of nonlinear systems with the linearized Jacobian matrix.

17. Bifurcations in a Competing Species Model
    • Duration: 31 min
    • Description: Analyze competing species using bifurcation and linearization techniques.

• Limit Cycles and Oscillations in Chemistry
  • Duration: 31 min
  • Description: Model oscillating reactions, expanding on equilibrium concepts in chemistry.

• All Sorts of Nonlinear Pendulums
  • Duration: 32 min
  • Description: Explore advanced techniques for solving nonlinear pendulum systems.

20. Periodic Forcing and How Chaos Occurs
    • Duration: 33 min
    • Description: Study the chaotic behavior of periodically forced systems.

21. Understanding Chaos with Iterated Functions
    • Duration: 31 min
    • Description: Simplify chaotic behavior through iterated functions and difference equations.

• Periods and Ordering of Iterated Functions
  • Duration: 32 min
  • Description: Examine the significance of fixed points and periodic points in differential equations.

23. Chaotic Itineraries in a Space of All Sequences
    • Duration: 33 min
    • Description: Analyze chaotic behavior outside the real line through complex algorithms.

24. Conquering Chaos-Mandelbrot and Julia Sets
    • Duration: 33 min
    • Description: Study fractals in the complex plane and their connection to differential equations.