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Differential Equations: 2x2 Systems

In order to understand most phenomena in the world, we need to understand not just single equations, but systems of differential equations. In this course, we start with 2x2 systems.

Differential Equations: 2x2 Systems
This course is archived
Estimated 10 weeks
2–5 hours per week
Instructor-led on a course schedule
Optional upgrade available

About this course

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Differential equations are the language of the models we use to describe the world around us. Most phenomena require not a single differential equation, but a system of coupled differential equations. In this course, we will develop the mathematical toolset needed to understand 2x2 systems of first order linear and nonlinear differential equations. We will use 2x2 systems and matrices to model:

  • predator-prey populations in an ecosystem,
  • competition for tourism between two states,
  • the temperature profile of a soft boiling egg,
  • automobile suspensions for a smooth ride,
  • pendulums, and
  • RLC circuits that tune to specific frequencies.

The five modules in this seriesare being offered as an XSeries on edX. Please visit the Differential EquationsXSeries Program Page to learn more and to enroll in the modules.

  • Wolf photo by Arne von Brill on Flickr (CC BY 2.0)
  • Rabbit photo by Marit & Toomas Hinnosaar on Flickr (CC BY 2.0)


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At a glance

  • Institution: MITx
  • Subject: Math
  • Level: Intermediate
  • Prerequisites:

    18.031x Introduction to Differential Equations (Scalar equations)

What you'll learn

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  • How to model real world problems by 2x2 systems of differential equations
  • How to use matrix methods to solve homogeneous systems of 2 first order linear differential equations
  • How to use graphical methods to understand the qualitative behavior of linear and nonlinear systems, and how to apply linear approximation to nonlinear (autonomous) 2x2 systems

Unit 1: Linear 2x2 systems
1. Introduction to systems of differential equations
2. Solving 2x2 homogeneous linear systems of differential equations
3. Complex eigenvalues, phase portraits, and energy
4. The trace-determinant plane and stability

Unit 2: Nonlinear 2x2 systems

5. Linear approximation of autonomous systems
6. Stability of autonomous systems
7. Nonlinear pendulum

About the instructors

Frequently Asked Questions

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Is 18.03Lx a prerequisite for this course?

No. The only prerequisite is 18.031x. 18.03Lx treats the special topic of the Laplace transform, and is aimed at more advanced and experienced students.

Who can take this course?

Unfortunately, learners residing in one or more of the following countries or regions will not be able to register for this course: Iran, Cuba and the Crimea region of Ukraine. While edX has sought licenses from the U.S. Office of Foreign Assets Control (OFAC) to offer our courses to learners in these countries and regions, the licenses we have received are not broad enough to allow us to offer this course in all locations. edX truly regrets that U.S. sanctions prevent us from offering all of our courses to everyone, no matter where they live.

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