Differential equations are the language of the models that we use to describe the world around us. In this series, we will explore temperature, spring systems, circuits, population growth, biological cell motion, and much more to illustrate how differential equations can be used to model nearly everything.
We will develop the mathematical tools needed to solve linear differential equations, understand 2x2 systems of first-order linear and nonlinear differential equations. We will use linear algebra to solve systems of more than two differential equations. As well as, explore the use of Fourier series to analyze the behavior of and solve ordinary differential equations (ODEs) and separable partial differential equations (PDEs). In the last course of the series, we will consider frequency domain and Laplace transform to help us appreciate their effects on mechanical and electrical systems.
"Wonderful course on differential equations. The teachers provide a nice computational tool to depict the dynamics of solving the equations, which is very useful for students to grasp the key ideas and concepts.” - Jiting (completed this course, spending 10 hours a week on it and found the course difficulty to be medium)
"Interesting course. Lectures, homeworks and review exercises of any part are really well setup. One of the best MOOC on topic of differential equations." - Gaetano (completed this course, spending 4 hours a week on it and found the course difficulty to be medium)
"Another excellent course from MIT. The lecture videos are excellent and so are the exercises. This course also has MATLAB based exercises which is wonderful. The problem sets are excellent and so are the staff and the community teaching assistants who are always there to help any time." - Dna47a (completed this course, spending 8 hours a week on it and found the course difficulty to be medium)
Courses under this program: Course 1: Introduction to Differential Equations
Scientists and engineers understand the world through differential equations. You can too.
Course 2: 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.
Course 3: 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.
Course 4: Differential Equations: Fourier Series and Partial Differential Equations
Learn to use Fourier series to solve differential equations with periodic input signals and to solve boundary value problems involving the heat equation and wave equation.
Course 5: Transfer Functions and the Laplace Transform
An intro to the mysteries of the frequency domain and Laplace transform and how they're used to understand mechanical and electrical systems.
This course is about the Laplace Transform, a single very powerful tool for understanding the behavior of a wide range of mechanical and electrical systems: from helicopters to skyscrapers, from light bulbs to cell phones. This tool captures the behavior of the system and displays it in highly graphical form that is used every day by engineers to design complex systems.
This course is centered on the concept of the transfer function of a system. Also called the system function, the transfer function completely describes the response of a system to any input signal in a highly conceptual manner. This visualization occurs not in the time domain, where we normally observe behavior of systems, but rather in the “frequency domain.” We need a device for moving from the time domain to the frequency domain; this is the Laplace transform.
We will illustrate these principles using concrete mechanical and electrical systems such as tuned mass dampers and RLC circuits.
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:
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,
Differential equations are the mathematical language we use to describe the world around us. Many phenomena are not modeled by differential equations, but by partial differential equations depending on more than one independent variable. In this course, we will use Fourier series methods to solve ODEs and separable partial differential equations (PDEs). You will learn how to describe any periodic function using Fourier series, and will be able to use resonance and to determine the behavior of systems with periodic input signals that can be described in terms of Fourier series. This course will use MATLAB to assist computations.
In this course we will explore:
How to process noisy sound files
The way a beam bends in response to external forces
How to design of ovens to create strong but lightweight composites