# Differential Equations: Linear Algebra and NxN Systems of Differential Equations

Learn how to use linear algebra and MATLAB to solve large systems of differential equations.

## There is one session available:

After a course session ends, it will be archived.
Estimated 9 weeks
5–8 hours per week
Instructor-paced
Instructor-led on a course schedule

Differential equations are the mathematical language we use to describe the world around us. Most phenomena can be modeled not by single differential equations, but by systems of interacting differential equations. These systems may consist of many equations. In this course, we will learn how to use linear algebra to solve systems of more than 2 differential equations. We will also learn to use MATLAB to assist us.

We will use systems of equations and matrices to explore:

• The original page ranking systems used by Google,
• Balancing chemical reaction equations,
• Tuned mass dampers and other coupled oscillators,
• Threeor more species competing for resources in an ecosystem,
• The trajectory of a rider on a zipline.

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.

*Zipline photo by teanitiki on Flickr (CC BY-SA 2.0)

### At a glance

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

18.031x Introduction to Differential Equations (Scalar equations), 18.032x Differential Equations: 2x2 Systems (2x2 first order differential equations)

# What you'll learn

Skip What you'll learn

After this course, you will be able to

• Model and solve different real world phenomena with systems of differential equations.
• Find the dimension and a basis for a (finite dimensional) vector space.
• Formulate and solve eigenvalue and eigenvector problems.
• Use MATLAB to explore solutions to large systems of equations.

# Syllabus

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Unit 1: Linear Algebra

1. Solving linear systems: elimination and RREF
2. Nullspace, vector space
3. Column space, determinants, and inverses
4. eigenvalues, eigenvectors, and diagonalization

Unit 2: Systems of Differential Equations

1. Solving homogeneous NxN systems of differential equations
2. Matrix exponential and diagonalization
3. Decoupling and solving inhomogeneous systems of equations
4. Solving nonlinear systems with MATLAB