Mastering Differential Equations: The Visual Method

Course No. 1452
Professor Robert L. Devaney, Ph.D.
Boston University
Share This Course
4.2 out of 5
46 Reviews
78% of reviewers would recommend this product
Course No. 1452
Video Streaming Included Free

Course Overview

For centuries, differential equations have been the key to unlocking nature's deepest secrets. Over 300 years ago, Isaac Newton invented differential equations to understand the problem of motion, and he developed calculus in order to solve differential equations.

Since then, differential equations have been the essential tool for analyzing the process of change, whether in physics, engineering, biology, or any other field where it's important to predict how something behaves over time.

The pinnacle of a mathematics education, differential equations assume a basic knowledge of calculus, and they have traditionally required the rote memorization of a vast "cookbook" of formulas and specialized tricks needed to find explicit solutions. Even then, most problems involving differential equations had to be simplified, often in unrealistic ways; and a huge number of equations defied solution at all using these techniques.

But that was before computers revolutionized the field, extending the reach of differential equations into previously unexplored areas and allowing solutions to be approximated and displayed in easy-to-grasp computer graphics. For the first time, a method exists that can start a committed learner on the road to mastering this beautiful application of the ideas and techniques of calculus.

Mastering Differential Equations: The Visual Method takes you on this amazing mathematical journey in 24 intellectually stimulating and visually engaging half-hour lectures taught by a pioneer of the visual approach, Professor Robert L. Devaney of Boston University, coauthor of one of the most widely used textbooks on ordinary differential equations.

Differential Equations without Drudgery

A firm believer that there is no excuse for drudgery in a subject as fascinating as differential equations, Professor Devaney draws on the power of the computer to explore solutions visually. Throughout these graphics-intensive lectures, you investigate the geometric behavior of differential equations, seeing how the computer can calculate approximate solutions with as much precision as needed. And you may be surprised to learn how easily you can calculate and display approximate solutions yourself, even using nothing more than an ordinary spreadsheet. Best of all, the visual method means that unrealistic simplifications need not be applied to a problem.

Among those who will benefit from the exciting approach in Mastering Differential Equations are

  • college students currently enrolled in a differential equations course, who want the enriching perspective of a leader in the visual approach to the subject;
  • anyone who has completed calculus, is ready to take the next step, and is eager to see how the tools of calculus are applied to give startling insights into nature;
  • those who took differential equations in the past and would like a refresher course, especially one that shows today's revolutionary new tools for demystifying and extending the reach of the subject;
  • anyone who finds math exciting, is up for a challenge, and wants a new window into the elegantly simple structure at the heart of nature's most complex phenomena.

Beautiful Ideas plus Amazing Applications

Differential equations involve velocity, acceleration, growth rates, and other quantities that can be interpreted by derivatives, which are a fundamental concept of calculus. Often expressed with utmost simplicity and mathematical elegance, differential equations underlie some of nature's most intriguing phenomena:

  • The first and most famous differential equation is Isaac Newton's second law of motion (F = ma), which relates force, mass, and acceleration, allowing the velocity and position of an accelerating object to be determined at any point in time.
  • The Lorenz differential equation for modeling weather describes the behavior of the atmosphere in terms of a single fluid particle, showing how nature's inherent chaos can be modeled with surprisingly simple mathematics.
  • Differential equations have been used to model the catastrophic behavior of the Tacoma Narrows Bridge, which famously collapsed in 1940, and London's Millennium Bridge, which appeared headed for the same fate before it was quickly closed for modifications in 2000.
  • The precipitous drop in the North Atlantic haddock population can be understood as the bifurcation point in a differential equation, in which a slight change in one parameter—the harvesting rate—produces a drastic effect on population growth.

These and countless other applications illustrate the unrivaled ability of differential equations to stop time and sharpen our view into the past and future—a power that has grown enormously with advances in computer technology, as you explore in depth in this course.

A Taste of 21st-Century Mathematics

Each of the four sections of Mastering Differential Equations begins with a phenomenon that can be modeled with differential equations. As you probe deeper, fundamental ideas (the derivative, integral, and vector field) and other relevant tools from calculus are introduced, along with new mathematics, including four lectures on linear algebra and five lectures on chaos theory.

In the first section, you cover first-order differential equations, which involve only the first derivative of the missing function being sought as a solution. When possible, you solve the equations analytically, while making use of a wide range of visual tools, including slope fields, phase lines, and bifurcation diagrams. You also learn how computers use a simple algorithm to generate approximate solutions—and how these techniques can sometimes fail, often due to chaos.

In the second section, you turn to second-order differential equations—those that involve both the first and second derivatives. Using the mass-spring system from physics as a model, you learn that solutions are relatively straightforward as long as the mass-spring system is not forced. But when periodic forcing occurs, much more complicated behaviors arise.

In the third section, you focus on systems of differential equations, starting with linear systems and the techniques of linear algebra, which are pivotal for solving such problems. Then you shift to nonlinear systems, investigating competing species, oscillating chemical reactions, and the Lorenz system of weather modeling—which led to the famous "butterfly effect," one of the ideas that spawned chaos theory.

The final section goes deeper into chaos theory, introducing you to the cutting-edge field of dynamical systems, whose development has exploded, thanks to the rise of visual methods. Here you focus on iterated functions, also known as difference equations. Using the logistic population model from biology, you learn to analyze and understand the sudden appearance of chaos. Then you move onto the complex plane to graph the visually stunning chaos that emerges in such fractal forms as the Mandelbrot set, taking you into realms of cutting-edge mathematics.

The winner of many teaching honors, including the prestigious Deborah and Franklin Tepper Haimo Award for Distinguished University Teaching from the Mathematical Association of America, Professor Devaney is especially excited about the modern mathematics that he brings to this course. Just a few years ago, he notes, students studying differential equations seldom ventured beyond 18th-century mathematics. But Mastering Differential Equations guides you into the 21st century, showing how this deceptively simple tool—the differential equation—continues to give surprising and spectacular insights into both the world of mathematics and the workings of the universe.

Hide Full Description
24 lectures
 |  Average 31 minutes each
  • 1
    What Is a Differential Equation?
    A differential equation involves velocities or rates of change. More precisely, it is an equation for a missing mathematical function (or functions) in terms of the derivatives of that function. Starting with simple examples presented graphically, see why differential equations are one of the most powerful tools in mathematics. x
  • 2
    A Limited-Growth Population Model
    Using a limited-growth population model (also known as a logistic growth model), investigate several ways to visualize solutions to autonomous first-order differential equations—those that involve only the first derivative and that do not depend on time. Plot slope-field and solution graphs, and learn about a pictorial tool called a phase line. x
  • 3
    Classification of Equilibrium Points
    Explore the concepts of source, sink, and node. These are the three types of equilibrium solutions to differential equations, which govern the behavior of nearby solutions on a graph. Then turn to the existence and uniqueness theorem, perhaps the most important theorem regarding first-order differential equations. x
  • 4
    Bifurcations—Drastic Changes in Solutions
    Sometimes tiny differences in the value of a parameter in a differential equation can lead to drastic changes in the behavior of solutions—a phenomenon called bifurcation. Probe an example involving the harvesting rate of fish, finding the bifurcation point at which fish stocks suddenly collapse. x
  • 5
    Methods for Finding Explicit Solutions
    Turning from the qualitative computer-based approach, try your hand at the standard methods of solving differential equations, specifically those for linear and separable first-order equations. Professor Devaney first reviews integration—the technique from calculus used to solve the examples, including one problem illustrating Newton's law of cooling. x
  • 6
    How Computers Solve Differential Equations
    Computers have revolutionized the solution of differential equations. But how do they do it? Learn one simple approach, Euler's method, which allows a very straightforward approximation of solutions. Test it using one of the most surprisingly powerful tools for analyzing differential equations: spreadsheets. x
  • 7
    Systems of Equations—A Predator-Prey System
    Embark on the second part of the course: systems of differential equations. These are collections of two or more differential equations for missing functions. An intriguing example is the fluctuating population of foxes and rabbits in a predator-prey relationship, each represented by a differential equation. x
  • 8
    Second-Order Equations—The Mass-Spring System
    Advancing to second-order differential equations (those involving both the first and second derivatives), examine a mass-spring system, also known as a harmonic oscillator. Taking three different views of the system, watch its actual motion, its solutions in the phase plane, and the graph of its changing position and velocity. x
  • 9
    Damped and Undamped Harmonic Oscillators
    Consider cases of a spring with no or very little friction. In solving these differential equations, encounter one of the most beautiful and important formulas in all of mathematics, Euler's formula, which shows the deep link between complex exponential functions and trigonometric functions. x
  • 10
    Beating Modes and Resonance of Oscillators
    Analyze what happens when force is applied to a spring in a periodic fashion. The resulting motions are very different depending on the relationship of the natural frequency and the forcing frequency. When these are the same, disaster strikes—a phenomenon that may have contributed to the famous collapse of the Tacoma Narrows Bridge. x
  • 11
    Linear Systems of Differential Equations
    Begin a series of lectures on linear systems of differential equations by delving into linear algebra, which provides tools for solving these problems. Review vector notation, matrix arithmetic, the concept of the determinant, and the conditions under which equilibrium solutions arise. x
  • 12
    An Excursion into Linear Algebra
    Explore more ideas from linear algebra, learning about eigenvalues and eigenvectors, which are the key to finding straight-line solutions for linear systems of differential equations. From these special solutions, all possible solutions can be generated for any given initial conditions. x
  • 13
    Visualizing Complex and Zero Eigenvalues
    Professor Devaney summarizes the steps for solving linear systems of differential equations, pointing out that complex eigenvalues are one possibility. Discover that in this case Euler's formula is used, which yields solutions that depend on both exponential and trigonometric functions. x
  • 14
    Summarizing All Possible Linear Solutions
    Turn to the special cases of repeated eigenvalues and zero eigenvalues. Then end this part of the course with a computer visualization of all possible types of phase planes for linear systems, seeing their connection to the bifurcation diagrams from Lecture 4. x
  • 15
    Nonlinear Systems Viewed Globally—Nullclines
    Most applications of differential equations arise in nonlinear systems. Begin your study of these challenging problems with a nonlinear model of a predator-prey relationship. Learn to use an analytical tool called the nullcline to get a global picture of the behavior of solutions in such systems. x
  • 16
    Nonlinear Systems near Equilibria—Linearization
    Experiment with another tool for coping with nonlinear systems: linearization. Given an equilibrium point for a nonlinear system, it's possible to approximate the behavior of nearby solutions by dropping the nonlinear terms and considering the corresponding linearized system, which involves an expression called the Jacobian matrix. x
  • 17
    Bifurcations in a Competing Species Model
    Combine linearization and nullclines to analyze what happens when two species compete. The resulting system of differential equations depends on several different parameters, yielding many possible outcomes—from rapid extinction of one species to a coexistence equilibrium for both. As the parameters change, bifurcations arise. x
  • 18
    Limit Cycles and Oscillations in Chemistry
    Use nullclines and linearization to investigate a startling phenomenon in chemistry. Before the 1950s, it was thought that all chemical reactions tended to equilibrium. But the Russian chemist Boris Belousov discovered a reaction that oscillated for hours. Your analysis shows how differential equations can model this process. x
  • 19
    All Sorts of Nonlinear Pendulums
    Focusing on the nonlinear behavior of a pendulum, learn new ways to deal with nonlinear systems of differential equations. These include Hamiltonian and Lyapunov functions. A Hamiltonian function remains constant along all solutions of special differential equations, while a Lyapunov function decreases along all solutions. x
  • 20
    Periodic Forcing and How Chaos Occurs
    Study the behavior of a periodically forced nonlinear pendulum to see how tiny changes in the initial position lead to radically different outcomes. To understand this chaotic behavior, turn to the Lorenz equation from meteorology, which was the first system of differential equations to exhibit chaos. x
  • 21
    Understanding Chaos with Iterated Functions
    Mathematicians understand chaotic behavior in certain differential equations by reducing them to an iterated function (also known as a difference equation). Try several examples using a spreadsheet. Then delve deeper into the subject by applying difference equations to the discrete logistic population model. x
  • 22
    Periods and Ordering of Iterated Functions
    Continuing with the discrete logistic population model, notice that fixed and periodic points play the role in difference equations that equilibrium points play in differential equations. Also investigate Sharkovsky's theorem from 1964, a result that heralded the first use of the word "chaos" in the science literature. x
  • 23
    Chaotic Itineraries in a Space of All Sequences
    How do mathematicians understand chaotic behavior? Starting with a simple function that is behaving chaotically, move off the real line and onto what at first appears to be a much more complicated space, but one that is an ideal setting for analyzing chaos. x
  • 24
    Conquering Chaos—Mandelbrot and Julia Sets
    What is the big picture of chaos that is now emerging? Center your investigation on the complex plane, where iterated functions produce shapes called fractals, including the Mandelbrot and Julia sets. Close by considering how far you've come—from the dawn of differential equations in the 17th century to fractals and beyond. x

Lecture Titles

Clone Content from Your Professor tab

What's Included

What Does Each Format Include?

Video DVD
Instant Video Includes:
  • Download 24 video lectures to your computer or mobile app
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps
Video DVD
DVD Includes:
  • 24 lectures on 4 DVDs
  • 320-page printed course guidebook
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps

What Does The Course Guidebook Include?

Video DVD
Course Guidebook Details:
  • 320-page printed course guidebook
  • Charts, diagrams & equations
  • Practice problems & solutions
  • Suggested readings

Enjoy This Course On-the-Go with Our Mobile Apps!*

  • App store App store iPhone + iPad
  • Google Play Google Play Android Devices
  • Kindle Fire Kindle Fire Kindle Fire Tablet + Firephone
*Courses can be streamed from anywhere you have an internet connection. Standard carrier data rates may apply in areas that do not have wifi connections pursuant to your carrier contract.

Your professor

Robert L. Devaney

About Your Professor

Robert L. Devaney, Ph.D.
Boston University
Dr. Robert L. Devaney is Professor of Mathematics at Boston University. He earned his undergraduate degree from the College of the Holy Cross and his Ph.D. from the University of California, Berkeley. His main area of research is dynamical systems, including chaos. Professor Devaney's teaching has been recognized with many awards, including the Feld Family Professor of Teaching Excellence, the Scholar/Teacher of the Year,...
Learn More About This Professor
Also By This Professor


Mastering Differential Equations: The Visual Method is rated 4.2 out of 5 by 46.
Rated 5 out of 5 by from Excellent Course! I bought this course a while back but finally got into it. I preceded it with the Understanding Calculus I & II Great Courses. One could take the video lectures on its own and would benefit from it. On the other hand I wanted to gain a solid understanding of the Course. Just like any university course, I did the homework at the end of the lesson. It was very beneficial and the problems really helped. I purchased the text book "Differential Equations, 4th Edition". It was a great supplement to the guide book. To further apply my knowledge I used a Wolfram Mathematica subscription to reproduce the visuals in the lectures and as an aid in the home work. Just like college, instruction, textbook, homework and lab leads to a full understanding.
Date published: 2020-09-02
Rated 5 out of 5 by from Amazing course I´m studying this course. It´s great. A different approach to understand differential equations ¿Is there a similar method to study partial differential equations?
Date published: 2020-06-23
Rated 5 out of 5 by from Great Instructor. Have not received the DVD yet but I am streaming the course. The instructor so far has made the material understandable! I passed Calculus II in college, and want to learn more on my own. I know it doesn't have to do with the course itself, but FedEx service is poor..
Date published: 2020-05-11
Rated 5 out of 5 by from Great perspective This is simply a great course. When I first took differential equations the emphasis was on solving the equation. The approach this course takes helps you understand what the equation is telling you. It really helps to understand the importance and value of differential equations.
Date published: 2020-02-27
Rated 4 out of 5 by from Key Word: Mastering. I bought this course several years ago. I finally watched the first 7 lectures. The professor communicates very well in the video classroom. As much as I can tell, he knows his subject well, and his teaching methods are fine for anyone who has already taken an introductory differential equations course. I have not. I have spent the previous 2 years reviewing calculus during my 8 hours per week of train commuting, and pretty much everything is going right over my head. I did not need long to realize that the prerequisite for this course is an introductory differential equations course. Everyone else will be lost very quickly. I would recommend this product to a friend who has taken an introductory differential equations course course. I would not recommend this product to a friend who has not.
Date published: 2019-11-17
Rated 5 out of 5 by from Interesting Subject Matter I loved the graphical nature of this course. I took differential equations in College when I was a senior. I was taught to memorize the different types of equations and the various types of solutions. We did not have computers back in 1980. The PC came out in 1982 and was severely limited in what it could do and cost about $3,000. Today a person can buy a computer for a $150 and solve their differential equation by typing in the equation and pushing a button online for free. They can also use Excel to solve the equation. The graphical nature of this course also shows exactly what is happening in picture format with the solutions. Something I had no idea about back in 1980. Twenty years after I graduated college my niece had differential equations as a senior in high school. I was amazed at this. She learned how to graph the differential equations. Now my sister is taking advanced math courses and they are teaching her how to integrate 7th order equations. Something that is needed to learn the future differential equations. I think they are using string theory with many extra dimensions for space and time and the reason for this higher order equations. They search for kids that can solve these equations so that they can research the new physics. A differential equation is an equation that has been differentiated and the person has to work the equation backwards to get at the general solution of a normal equation. I did not know this back in college back in 1980. My fellow students used to ask the professor what these equations were used for and they would never tell us. Now I know from taking this class. They are used to solve many different types of issues like predator/prey models to track populations of animals that may go extinct for example to set regulations on the amount of animals that may be hunted or fished to heavily.
Date published: 2019-10-07
Rated 4 out of 5 by from Supplemental DiffyQ fundamentals suggested I suggest viewing online lessons and getting a firmer understanding of the mechanics of the specific types of differential equations discussed in the MDE Course. Once I did so I was able to appreciate the approach Professor Delaney was taking. Perhaps this should have been 36 lectures not 24, and covered the traditional diffq techniques in a similar way to the other math courses in the series.
Date published: 2019-03-25
Rated 4 out of 5 by from Oops- was not a different course. I already had this course but thought this was a new one. Now I have two courses the same. Wish your ordering had a history check to prevent this sort of thing.
Date published: 2018-12-24
  • y_2020, m_11, d_24, h_16
  • bvseo_bulk, prod_bvrr, vn_bulk_3.0.12
  • cp_1, bvpage1
  • co_hasreviews, tv_4, tr_42
  • loc_en_US, sid_1452, prod, sort_[SortEntry(order=SUBMISSION_TIME, direction=DESCENDING)]
  • clientName_teachco
  • bvseo_sdk, p_sdk, 3.2.0
  • CLOUD, getContent, 9.01ms

Questions & Answers

Customers Who Bought This Course Also Bought

Buy together as a Set
Save Up To $21.05
Choose a Set Format