Summary

Differential equations are equations that include a function and one or more of its derivatives. They were discovered by Sir Isaac Newton and Gottfried Wilhelm Leibniz, the co-­discoverers of calculus. Differential equations are important in mathematics, engineering, and many scientific fields, where they are used to describe how things change.

Overview

From the point of view of physics, perhaps the most important contribution of Sir Isaac Newton was the realization that many laws of nature are expressed by relations between functions and their derivatives. For example, in mechanics, if a body is moving under the gravitational influence of others, the position determines the force acting on it and, by Newton’s law of motion, dividing this force by the mass of the body, one can determine the acceleration.

Since the acceleration is the second derivative of the position with respect to time, one finds that there should be a relationship between the position of a body and its second derivative with respect to time. Conversely, every motion described by functions satisfying the differential equation is possible for the system if one puts it in the appropriate initial state. Similar examples can be found in many physical sciences. For example, the rate of cooling of an object—­the rate of change of its temperature—­is very often proportional to the difference of temperatures between the body and its surrounding media.

Such relationships between functions and their derivatives are called “differential equations.” Given a differential equation, one may attempt to find all the possible functions that satisfy the equation. This is called “finding the general solution.” If one succeeds in doing that, by reading out the formulas one may discover all possible motions that the system may experience. This may be useful if one wants to find some particular type of motion with desirable properties. For example, if one has an explicit general solution for the pendulum, one may inquire about whether there is a motion in which the pendulum rotates twice and stops.

Unfortunately, this procedure of finding explicit solutions is very difficult and, for some systems, even impossible. Many times one must settle for simpler problems. For example, one may prescribe the initial position and velocity of a space ship and ask where it will be one year from now. This is called the “initial value problem.”

One important advantage of the differential equations method compared with other methods of encapsulating the laws of nature is that it allows its user to obtain approximate solutions systematically. One can, for example, use numerical methods, or one can systematically find out what the corrections are, starting from a simpler model. This is called a perturbation analysis.

Perhaps the biggest triumph of the method of perturbation analysis is in solar system studies. If the planets had very small masses, the solar system could be understood using explicit formulas and one could derive Kepler’s laws. In the time of Newton, it was known that these laws did not fit the observations exactly. Newton and others, notably Pierre-­Simon de Laplace, found that most of these observations could be accounted for by the fact that the planets are massive. Even if they could not find exact formulas for the motions of the planets, they could find formulas that, even if approximate, were of comparable or better accuracy than the experimental observations of the time. As the techniques to obtain data were refined, it became necessary to work harder at improving the accuracy of the approximate formulas.

It was believed for a long time that this procedure of deriving more and more approximate solutions could be carried out to any degree of approximation. Nevertheless, in the last decades of the nineteenth century, Henri Poincaré started a systematic study of perturbation methods and discovered that some of them have intrinsic limitations. He also proposed a new way of studying differential equations, now called “the geometric approach.” The basic idea is that many of the questions one asks about differential equations are really geometric questions.

For example, when one asks about an orbit going from the Earth to the Moon and back, one is really asking about a line passing through regions in space. Even if one were to succeed in finding explicit formulas for the motions, it would be necessary to analyze the formulas to verify whether such orbits are possible. It then becomes natural to devise methods of reasoning that work with geometric objects using geometric arguments without making use of the crutch of deriving explicit formulas, whose geometric interpretation has to be worked out afterward.

One result along this line—­derived before Poincaré—­is the continuity with respect to initial conditions. For many differential equations, it is possible to show that any initial condition determines uniquely where one will be one unit of time later. Moreover, if one makes a small error in the initial condition, the corresponding error in the position one unit of time later is also small. This propagated error can be made as small as one wishes by ensuring that the error in the initial conditions is small.

For many differential equations whose defining functions admit derivatives of high order, one can get considerably sharper results. The results derived from the geometric program have had an enormous influence not only in applications but also in mathematics. Many new disciplines, such as topology, were created to serve as tools for the study of differential equations and then took on a life of their own. One important development has been the availability of digital computers. It is possible to write algorithms that produce very approximate solutions of the ordinary differential equations. In fact, one of the first problems that was tackled by computers was the production of “artillery tables”—­which solve differential equations that model the motion of a shell in the atmosphere subject to gravitation and friction.

The influence of computers has been very profound. For concrete applications in which the goal is to compute an actual orbit whose initial conditions are known, they are now the tool of choice. Perhaps more important, through judicious simulation of key cases, it is possible to develop intuitions that help solve the problem and lead to the understanding of new phenomena. Graphical representations that are easy to grasp have been developed, and the intuitions obtained through these graphical simulations are particularly helpful when used together with the mathematical results coming from the geometric approach. There are already several commercially available programs for the visual exploration of differential equations.

Applications

Differential equations arose from the needs of classical mechanics, but they are fundamental tools for almost all branches of physical sciences and, more tentatively, for biology and economics.

In application to mechanics, the success has been spectacular. Almost all features of the solar system have been accounted for by differential equations, with the exception of a few notable examples including a convincing explanation of the rings of Saturn, the fact that the Moon always faces the Earth, and that Mercury rotates exactly three times around its axis every two revolutions around the sun. It is also possible in a routine way to compute the motion of artificial satellites and of the Moon in such a way that the effects of all planets are included so as to get a precision of about a meter for the position of the Moon over a century. More important, it is possible to find orbits that are immune to disturbances or that use them to get several effects.

Differential equations gave rise to many mechanical inventions that dominated the science and technology until the beginning of the twentieth century. When mechanical devices were replaced by electric and electronic ones, differential equations remained the method of choice. By using very simple rules, it is easy to derive differential equations that model circuits.

The difference of voltage across a capacitor is proportional to the charge that it stores; the derivative with respect to time of this charge is the current flowing to the capacitor. The difference of voltage across a resistor is proportional to the current flowing through it (Ohm’s law). For an electronic device such as a diode, the voltage as function of the current is a complicated nonlinear function. The voltage attributable to self-­induction is proportional to the rate of change of the current. For some more complicated devices, such as transistors or vacuum tubes, the current flowing across a pair of legs is a function of the voltage applied to them as well as the voltage of a third leg. Such equations are at the basis of all electronic applications. For example, it is possible to understand how to build circuits that will keep oscillating with a fixed frequency. Such circuits are found in many useful devices, such as radio transmitters and receptors, television sets and computers.

Another important application of differential equations is in the kinetics of chemical reactions. The rate of change of the concentration of chemicals in a reactor tank is proportional to the number of reactions that take place. Those, in turn, are proportional to the number of collisions of appropriate molecules, which in turn depends in a simple way on the concentrations. By using differential equations, it is possible to predict conditions in which the end product will be a useful material rather than a useless waste. It is also possible to predict regimes which stay safely away from dangerous behaviors such as explosions.

Besides the direct applications of differential equations to physics, many problems in mathematics—­frequently arising from physics—­can be reduced to differential equations. A particularly important one is the use of the method of separation of variables in partial differential equations. Other problems that frequently lead to differential equations are variational problems in which one tries to determine functions that are optional in a certain sense.

Context

Differential equations appeared with calculus in the hands of Newton as a tool for mechanics, and quickly developed into the method of choice for modeling physical systems. A drastic revolution took place at the end of the nineteenth century, when several mathematicians realized that it was advantageous to think of differential equations in geometric terms. This geometric program remained dormant for a long time. In the West, it was used mainly by mechanical engineers and those who studied celestial mechanics, whereas in the East it was developed mainly by electrical engineers. In the 1960s, it again caught the attention of mathematicians who, in the meantime, had developed many disciplines—­such as topology—­that had made precise many intuitions. Important leaders in this revival were the mathematicians Stephen Smale, Jack K. Hale, and Jürgen Moser in the West and Andrey N. Kolmogorov, Vladimir I. Arnold, and Yakov Sinai in the East.

One important tool that made progress quicker and easier was the increasing availability of computers. The graphic displays made it easy to obtain a geometric intuition of phenomena and to test conjectures. Computers were also the only means of obtaining quantitative predictions that were useful in applied problems.

From the late 1970s, the fact that differential equations could solve physical problems that had been unsolved for a long time was increasingly recognized by physicists. A notable turning point was the realization by Mitchell Feigenbaun that renormalization group methods could produce quantitative results for chaotic systems. From that time on, the field has experienced an explosive growth, and one can now find scientists in many disciplines (mathematics, physics, engineering, and even biology) making important contributions by using differential equations. Besides the specialized journals devoted specifically to differential equations, one can now find important results in many journals devoted to chemistry, physics, or biology.

Further Reading