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)

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
The motion of a violin string
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.
Violinist photo by user: DeshaCAM. Copyright © 2018 Adobe Systems Incorporated. Used with permission.

How do populations grow? How do viruses spread? What is the trajectory of a glider? Many real-life problems can be described and solved by mathematical models.
This course will introduce you to the modelling cycle which includes: analyzing a problem, formulating it as a mathematical model, calculating solutions and validating your results.
All models are (systems of) ordinary differential equations, and you will learn more about those by watching videos and reading short texts, and more importantly, by completing well-crafted exercises.
You will learn how to implement Euler's method in a (Python) program, and finally, you will learn how to write about your findings in a scientific way (with LaTeX).
In the verified track of this course you will additionally:
Consolidate the new theoretical skills with graded problem sets about five real-life applications.
Work on your own modelling project (individually or in a team). Because mathematical modelling is only learned by doing it yourself, you complete your own modelling project on a self-defined real-life problem. You will be guided through the project by completing a list of smaller tasks.
This course is aimed at Bachelor students from Mathematics, Engineering and Science disciplines.
The course is for anyone who would to use mathematical modelling for solving real world problems, including business owners, researchers and students.

The course is practice-oriented. It is supplemented with many problems aimed at assisting the understanding of lecture materials.
Each problem, in turn, is supplemented with a detailed solution.
The topics covered:
1. Complex algebra, complex differentiation, simple conformal mappings.
2. Taylor and Laurent expansion.
3. Residue theory. Integration of contour and real integrals with the help of residues.
4. Multivalued functions and regular branches
5. Analytic continuation and Riemann surfaces.
6. Integrals with multivalued functions.
The course includes two tracks.
The free track allows the learner to access all the materials from the course.
The "verified certificate" track allows the learner to
1. access additional non-trivial problems from the course.
2. access the detailed solutions to all the problems inside the course at the end of each week.
3. get an official certificate from the university on completion of the course.

Introduction to linear optimization, duality and the simplex algorithm.

Linear algebra is at the core of all of modern mathematics, and is used everywhere from statistics and data science, to economics, physics and electrical engineering. However, learning the subject is not principally about acquiring computational ability, but is more a matter of fluency in its language and theory.
In this course, we will start with systems of linear equations, and connect them to vectors and vector spaces, matrices, and linear transformations. We will be emphasizing the vocabulary throughout, so that students become comfortable working with the different aspects.
We will then introduce matrix and vector operations such as matrix multiplication and inverses, paying particular attention to their underlying purposes. Students will learn not just how to calculate them, but also why they work the way that they do.
We willdiscuss the key concepts of basis and dimension, which form the foundation for many of the more advanced concepts of linear algebra.
The last chapter concerns inner products, which allow us to use linear algebra for approximating solutions; we will see how this allows for applications ranging from statistics and linear regression to digital audio.

In this course, we go beyond the calculus textbook, working with practitioners in social, life and physical sciences to understand how calculus and mathematical models play a role in their work.
Through a series of case studies, you’ll learn:
How standardized test makers use functions to analyze the difficulty of test questions;
How economists model interaction of price and demand using rates of change, in a historical case of subway ridership;
How an x-ray is different from a CT-scan, and what this has to do with integrals;
How biologists use differential equation models to predict when populations will experience dramatic changes, such as extinction or outbreaks;
How the Lotka-Volterra predator-prey model was created to answer a biological puzzle;
How statisticians use functions to model data, like income distributions, and how integrals measure chance;
How Einstein’s Energy Equation, E=mc2 is an approximation to a more complicated equation.
With real practitioners as your guide, you’ll explore these situations in a hands-on way: looking at data and graphs, writing equations, doing calculus computations, and making educated guesses and predictions.
This course provides a unique supplement to a course in single-variable calculus. Key topics include application of derivatives, integrals and differential equations, mathematical models and parameters.
This course is for anyone who has completed or is currently taking a single-variable calculus course (differential and integral), at the high school (AP or IB) or college/university level. You will need to be familiar with the basics of derivatives, integrals, and differential equations, as well as functions involving polynomials, exponentials, and logarithms.
This is a course to learn applications of calculus to other fields, and NOT a course to learn the basics of calculus. Whether you’re a student who has just finished an introductory Calculus course or a teacher looking for more authentic examples for your classroom, there is something for you to learn here, and we hope you’ll join us!

Matrix Algebra underlies many of the current tools for experimental design and the analysis of high-dimensional data. In this introductory online course in data analysis, we will use matrix algebra to represent the linear models that commonly used to model differences between experimental units. We perform statistical inference on these differences. Throughout the course we will use the R programming language to perform matrix operations.
Given the diversity in educational background of our students we have divided the series into seven parts. You can take the entire series or individual courses that interest you. If you are a statistician you should consider skipping the first two or three courses, similarly, if you are biologists you should consider skipping some of the introductory biology lectures. Note that the statistics and programming aspects of the class ramp up in difficulty relatively quickly across the first three courses. You will need to know some basic stats for this course. By the third course will be teaching advanced statistical concepts such as hierarchical models and by the fourth advanced software engineering skills, such as parallel computing and reproducible research concepts.
These courses make up two Professional Certificates and are self-paced:
Data Analysis for Life Sciences:
PH525.1x: Statistics and R for the Life Sciences
PH525.2x: Introduction to Linear Models and Matrix Algebra
PH525.3x: Statistical Inference and Modeling for High-throughput Experiments
PH525.4x: High-Dimensional Data Analysis
Genomics Data Analysis:
PH525.5x: Introduction to Bioconductor
PH525.6x: Case Studies in Functional Genomics
PH525.7x: Advanced Bioconductor
This class was supported in part by NIH grant R25GM114818.

Introduction to the mathematical concept of networks, and to two important optimization problems on networks: the transshipment problem and the shortest path problem. Short introduction to the modeling power of discrete optimization, with reference to classical problems. Introduction to the branch and bound algorithm, and the concept of cuts.

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)