Mastering Differential Equations: The Visual Method

Mastering Differential Equations: The Visual Method
Course Trailer
What Is a Differential Equation?
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....

32 min
A Limited-Growth Population Model
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....

30 min
Classification of Equilibrium Points
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....

31 min
Bifurcations-Drastic Changes in Solutions
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....

30 min
Methods for Finding Explicit Solutions
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....

31 min
How Computers Solve Differential Equations
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....

28 min
Systems of Equations-A Predator-Prey System
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....

30 min
Second-Order Equations-The Mass-Spring System
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....

30 min
Damped and Undamped Harmonic Oscillators
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....

33 min
Beating Modes and Resonance of Oscillators
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....

32 min
Linear Systems of Differential Equations
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....

30 min
An Excursion into Linear Algebra
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....

33 min
Visualizing Complex and Zero Eigenvalues
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....

32 min
Summarizing All Possible Linear Solutions
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....

32 min
Nonlinear Systems Viewed Globally-Nullclines
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....

32 min
Nonlinear Systems near Equilibria-Linearization
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....

31 min
Bifurcations in a Competing Species Model
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....

31 min
Limit Cycles and Oscillations in Chemistry
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....

31 min
All Sorts of Nonlinear Pendulums
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....

32 min
Periodic Forcing and How Chaos Occurs
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....

33 min
Understanding Chaos with Iterated Functions
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....

31 min
Periods and Ordering of Iterated Functions
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....

32 min
Chaotic Itineraries in a Space of All Sequences
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....

33 min
Conquering Chaos-Mandelbrot and Julia Sets
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....

33 min
Robert L. Devaney

The field of differential equations has changed remarkably because of computer graphics. It is now fascinating to see how solutions of these equations evolve visually, especially those that are chaotic.

ALMA MATER

University of California, Berkeley

INSTITUTION

Boston University

About Robert L. Devaney

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, and the Metcalf Award for Teaching Excellence, all from Boston University; and the Deborah and Franklin Tepper Haimo Award for Distinguished University Teaching from the Mathematical Association of America. In 2002 he received a National Science Foundation Director's Award for Distinguished Teaching Scholars, as well as the International Conference on Technology in Collegiate Mathematics Award for Excellence and Innovation with the Use of Technology in Collegiate Mathematics. In 2004 he was named the Carnegie/CASE Massachusetts Professor of the Year, and in 2009 he was inducted into the Massachusetts Mathematics Educators Hall of Fame. Since 1989 Professor Devaney has been director of the National Science Foundation's Dynamical Systems and Technology Project, leading to a wide array of computer programs for exploring dynamical systems. He also produced the Mandelbrot Set Explorer, an online, interactive series to introduce students at all levels to the mathematics behind the fractal images known as the Mandelbrot and Julia sets. In addition to writing many professional papers and books, Professor Devaney is the coauthor of Differential Equations, a textbook now in its 4th edition, which takes a fundamentally visual approach to solving ordinary differential equations.

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