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.