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Mastering Differential Equations: The Visual Method

Mastering Differential Equations: The Visual Method

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Mastering Differential Equations: The Visual Method

Course No. 1452
Professor Robert L. Devaney, Ph.D.
Boston University
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4.1 out of 5
29 Reviews
75% of reviewers would recommend this series
Course No. 1452
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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.

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24 lectures
 |  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

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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,...
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Mastering Differential Equations: The Visual Method is rated 4.1 out of 5 by 29.
Rated 5 out of 5 by from Surprisingly interesting This is definitely a deeper course for people who like maths. As someone who tends to think visually, I wish I had this course 40 years ago at university -- I'd have appreciated and understood DEs a lot better. The course starts out gently with simple 1-D linear DEs, 1st order then 2nd order. Then into linear systems (only 2-D though) of DEs, with a good explanation of Eigenvalues. The second half of the course ramps up the intensity with nonlinear systems and chaotic behavior. For me this second half was really mind stretching, particularly the analysis of chaos at different levels of abstraction.
Date published: 2017-12-23
Rated 5 out of 5 by from Thank you... The best from the greatest... Mathematics is all about having passion, and learning from the greatest is invaluable...
Date published: 2017-11-22
Rated 5 out of 5 by from I bought 2 weeks ago differential equations course. Due to the first lecture, i was amazed by prof. explanation of basic mathematical explanation for introduce and prepare this courses. The second and third lecture, i got very good information to approach differential equations. Now , i know how can look difference this information and my all vision further one more step. Thank you for this amazing course. I wish there will be more mathematical course of prof.
Date published: 2017-10-14
Rated 3 out of 5 by from Not for the faint of heart You definitely need a calculus background to understand the course. Pretty good, but I thought, a bit too much reciting details, such as talking through long derivations, where it was obvious what the slide showed. Would like to have heard a few more real-life examples.
Date published: 2017-07-10
Rated 1 out of 5 by from LACONIC This is not for someone who already is familiar with Differential Equations. I viewed the first lecture and decided it was not for me
Date published: 2017-06-28
Rated 5 out of 5 by from One word: Spectacular ! Be ready for a deep dive Warning: The course is not a casual listening, dumbed down product. The course covers material typical of a first-year engineering cursus. I ended up taking copious notes. A knowledge of differential and integral calculus is required. Dedication is also a must. The course title “Mastering Differential Equations” is not marketing hype. By far the most advanced math course from TGC and the most challenging, its use of a simulation software package, available for free on the web, sets this course in a class all its own. It allows a comprehensive exploration of the behavior of differential equation systems – a graph is instantly produced in response to change of parameter value or initial condition. This tool provides a high-level view of differential equations systems and a rich insight which was simply not available in the days where the only approach was to work out by hand the solutions. This course provides the additional explanations I wished my teachers had covered, explaining how each term contributes to the whole equation when creating a mathematical model. The course also shows dramatic changes resulting from a small change in a parameter value with numerous animations. I must start with the last lecture: the Mandelbrot fractal picture. Prof. Devaney exposes how powerful this representation is – well beyond just a pretty picture. Its complexity is mind-boggling. Sort of ending the show with a fireworks display. Definitely stepping out of 17th century maths and moving into the 20th century. The course is easy to follow, until lectures 12 through 14 on linear algebra. These ones took considerably more time to absorb than their 30-minute run time. They are well worth the effort. The connection between differential equation systems and linear algebra is explained at length, showing how to use linear algebra tools to solve the d.e. systems. The description of eigenvalues and eigenvectors is top notch, especially the use of the trace and determinant of parameters matrix. The animation showing the behavior of the solutions in response to parameter changes must be seen. Outstanding ! In lecture 19 the bar is raised further with the exposition of Hamilton and Lyapunov functions. My first exposition to this. Clearly explained. Then an introduction to the relatively recent studies of the chaotic behavior of differential equations. Lecture 23 is at a higher level still, near my saturation point. End-of-19th-century pure maths disconnected from physics. Symbolic dynamics was not easy to swallow. At times dry, fortunately the course benefits from reconnection with real-life situations, such as modelling fish population against the impact of over-fishing and showing how a trivial increase in fishing can cause the fish population to collapse. The mass-spring system is another real-world example. Chaotic behavior is also covered. The demonstration of Euler’s formula linking numbers e, i, π, 1 and 0 is different from what I had learned and was a nice touch. This would provide a formidable advantage for anyone interested in a technical field which uses such equations. The course could also help improve the teaching of calculus by expanding upon the practical meaning instead of merely telling about the rules. I sincerely hope for more courses in the same vein on other math topics in vector calculus such as residue calculus – can someone PLEASE explain this beast without the same tired old stock answer (the Laurent series) which provides absolutely NOTHING as to the insight that lead to developing this tool. Or Greene’s and Stokes’ theorems. And rings and groups theory – with tensors … Overall, great show, despite some shortcomings I am mentioning below. Lecture 6 was a disappointment, repeated in lecture 21. The Excel worksheets shown automatically self-adjust to the ∆t increment changes, adding or removing rows and updating the chart – one worksheet even has a slider control to change the ∆t increment. When the explanation began on how to use Excel, I eagerly waited for the part on how to make the worksheet respond automatically to increment changes. Alas, the explanation dies a centimeter past the starting line, only showing to re-select the relevant rows to update the chart manually when the ∆t increment is changed – instantly quashing my hope of learning how to develop a really useful mathematical tool. As if someone who has already acquired knowledge in calculus needed to be shown the basic, basic, basic charting function of Excel. Bleh ! An entire lecture on how to create self-adapting worksheets would have been a magnificent addition to the course. I feel teased and cheated. Just for that I am taking out one star from the rating. OK that’s too severe, I’ll take one third of a star. I have not found the Excel worksheets on the web as easily as I found the Java applet of the Differential Equations simulation software. It would save some time. Lecture 7 dealing with predator-prey population evolution has a minor flaw: the choice of a, b, c and d as constants is ill advised as it causes the last term in the equilibrium equations to read as dR which could be confused with the differential of R – a minor distraction. This also occurs in lecture 21. The lack of the spreadsheets and of the orbit diagramming tool is also somewhat of a letdown.
Date published: 2017-04-08
Rated 2 out of 5 by from Touchy feely, not very mathematical Full disclosure - I couldn't be bothered to finish this course and only did the first 5 lectures. Very disappointed. I was looking for a to do an in-depth review of the subject. I've got a BSEE and have done this sort of stuff 35-40 years ago but wanted to refresh my skills. That was not the course to accomplish that goal. The other reviews have discussed the software issue so I won't belabor that here other than to say that without his software I don't think the course is worth much. I program in Matlab all the time and sure, most of what he's showing _could_ be done in Matlab if you wanted to invest the time to make it happen, but that's a bunch of work. Anyway, overall I bought this course on a whim when a sale was on so I didn't spend full price but what I did spend is, at this time, pretty much wasted. Maybe I'll try again later but Great Courses is asking me to review so there you are...
Date published: 2017-03-22
Rated 5 out of 5 by from Excellent course with great explanations I am very pleased that TGC Co. got into this subject and provided this course. While it's great to do introductory courses and I've done them in a number of areas with which I have less familiarity, I am also looking for more substantial and in-depth courses in areas where I have some knowledge or familiarity. Prof. DuVaney does an excellent job of going through this topic with a combination of theory, explanation, and application. In addition, as the course title conveys, he does it with graphical explanations as well. Many of the methods were not available when I went to school because experimenting with numerical solutions to problems was not really available given the mainframe computers of the day. However, this work available on desktop and laptop computers allows for "exploration" and "illumination" of the equations, their results and implications. I highly value this course and intend on re-watching it so that I can further expand my understanding of the topic and discussion. I hope you're able to enjoy it, too!
Date published: 2016-12-21
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