The following topics are covered in the course: complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis and Laplace transforms.

Complex analysis is a basic tool with a great many practical applications to the solution of physical problems. It revolves around complex analytic functions—functions that have a complex derivative. Unlike calculus using real variables, the mere existence of a complex derivative has strong implications for the properties of the function. Applications reviewed in this class include harmonic functions, two dimensional fluid flow, easy methods for computing (seemingly) hard integrals, Laplace transforms, and Fourier transforms with applications to engineering and physics.

This course provides the fundamental computational toolbox for solving science and engineering problems. Topics include review of linear algebra, applications to networks, structures, estimation, finite difference and finite element solutions of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, the discrete Fourier transform, and convolution. We will also explore many topics in AI and machine learning throughout the course.

This 6-minute video lesson looks at 2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent.

This 11-minute video lesson continues to look at 2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent.

This 9-minute video lesson continues to look at 2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent.

This 11-minute video lesson continues to look at 2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent.

This 12-minute video lesson concludes the look at 2010 IIT JEE Paper 1 Problem 45 Circle Hyperbola Common Tangent.

This 13-minute video lesson looks at 2010 IIT JEE Paper 1 Problem 46 Circle Hyperbola Intersection.

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
Course Format
This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by Professor Arthur Mattuck.
Course Notes on every topic.
Practice Problems with Solutions.
Problem Solving Videos taught by experienced MIT Recitation Instructors.
Problem Sets to do on your own with Solutions to check your answers against when you're done.
A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.
A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development
Haynes Miller 
Jeremy Orloff 
Dr. John Lewis 
Arthur Mattuck

This 10-minute video lesson provides an introduction to 2nd order, linear, homogeneous differential equations with constant coefficients. [Differential Equations playlist: Lesson 13 of 45]

This 8-minute video lesson continues the discussion from the previous video and shows how to find the general solution. [Differential Equations playlist: Lesson 14 of 45]

This 6-minute video lesson continues the discussion from the previous videos and introduces some initial conditions to solve for the particular solution. [Differential Equations playlist: Lesson 15 of 45]