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.
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,
Three or more species competing for resources in an ecosystem,
The trajectory of a rider on a zipline.
The five modules in this series are being offered as an XSeries on edX. Please visit the Differential Equations XSeries Program Page to learn more and to enroll in the modules.
*Zipline photo by teanitiki on Flickr (CC BY-SA 2.0)
Unit 1: Linear Algebra
Solving linear systems: elimination and RREF
Nullspace, vector space
Column space, determinants, and inverses
eigenvalues, eigenvectors, and diagonalization
Unit 2: Systems of Differential Equations
Solving homogeneous NxN systems of differential equations
Matrix exponential and diagonalization
Decoupling and solving inhomogeneous systems of equations
Solving nonlinear systems with MATLAB
18.031x Introduction to Differential Equations (Scalar equations), 18.032x Differential Equations: 2x2 Systems (2x2 first order differential equations)
What are the prerequisites for this course?
We assume familiarity with both 18.031x Introduction to Differential Equations and 18.032x Differential Equations: 2x2 systems. Knowledge from 18.03Lx is not required to succeed in this course.