RESEARCH STARTER

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives, capturing how a particular quantity changes over time or in relation to other variables. They play a crucial role in various fields, including physics, engineering, biology, and economics, by allowing the modeling of dynamic systems. For instance, Newton's laws of motion can be expressed through differential equations that relate an object's position to its acceleration.

These equations can be classified into different types, such as ordinary and partial differential equations, and finding solutions is essential for understanding the behavior of systems. Solutions can be either exact or approximate, with numerical methods often employed for complex problems. One powerful approach is perturbation analysis, which provides systematic ways to approximate solutions when exact solutions are difficult to obtain.

Over the years, the study of differential equations has evolved from purely mechanical applications to include insights from geometric perspectives, enriching the understanding of their solutions. The advent of computers has further transformed the field, enabling simulations and visualizations that enhance comprehension and facilitate predictions in real-world applications, from celestial mechanics to chemical reaction kinetics. As such, differential equations remain fundamental in both theoretical and applied mathematics, reflecting the intricate relationships present in nature.

Full Article

  • Type of physical science: Mathematical methods
  • Field of study: Calculus

Many laws of physics are best expressed by prescribing relationships between a function describing the phenomenon and its rate of change. Once these laws are supplemented by other conditions, such as the value of the functions at a specified time, it becomes possible to find out what actually happens.

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, as described by Newton’s law of cooling.

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 spaceship and ask where it will be one year later. This is called the “initial value problem.”

One important advantage of the differential equation’s 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 usually 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 the tool of choice. Perhaps more importantly, 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 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 and other fields such as epidemiology and finance.

In application to mechanics, the success has been spectacular. Almost all features of the solar system have been accounted for (notable examples are the rings of Saturn, the tidal locking that ensures the Moon always faces the Earth, and the spin-orbit resonance that causes Mercury to rotate 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 importantly, 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 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 in 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 in 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 a 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 receivers, televisions, 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 that 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 optimal 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 continued to develop through the early twentieth century, although it gained broader prominence later with the rise of dynamical systems theory. In both the West and the East, it was used mainly by mechanical engineers and those who studied celestial mechanics, but in the East, it was also used 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 J. Moser in the West and A. N. Kolmogorov, V. I. Arnold, and Y. Sinai in the East.

One important tool that made progress quicker and easier was the increasing availability of computers. The graphics made it easy to obtain a geometric intuition of phenomena and to test conjectures. Computers greatly enhanced the ways of obtaining quantitative predictions that were useful in applied problems, complementing earlier analytical and approximate methods.

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 Feigenbaum that renormalization group methods could produce quantitative results for chaotic systems. From that time on, the field experienced explosive growth, and one can 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 find important results in many journals devoted to chemistry, physics, or biology.

Principal terms

DIFFERENTIAL EQUATION: a relationship between the derivatives of one or more functions and the functions themselves

GENERAL SOLUTION OF A DIFFERENTIAL EQUATION: a formula involving arbitrary constants such that, by assigning values to the constants, one gets all the solutions of the differential equation

ORDER OF THE DIFFERENTIAL EQUATION: the order of the derivative of the highest order appearing in the differential equation

PARAMETERS: variables that do not enter the differential equation but that correspond to magnitudes that can be set from the exterior

SOLUTION OF A DIFFERENTIAL EQUATION: a function or group of functions whose derivatives are related to the functions in the way prescribed by the differential equation


Bibliography

Alazard, Thomas. Analysis and Partial Differential Equations. Springer, 2024.

Arnold, V. I. Ordinary Differential Equations. Translated by Richard A. Silverman, MIT Press, 1973.

Boyce, William E., and Richard C. DiPrima. Elementary Differential Equations and Boundary Value Problems. 4th ed., Wiley, 1986.

Braun, Martin. Differential Equations and Their Applications. 2nd ed., Springer-Verlag, 1978.

“Differential Equations.” Math Is Fun, www.mathsisfun.com/calculus/differential-equations.html. Accessed 24 Apr. 2026.

“Differential Equations.” Paul’s Online Notes, 26 June 2023, tutorial.math.lamar.edu/classes/de/de.aspx. Accessed 24 Apr. 2026.

“Differential Equations.” Wolfram, reference.wolfram.com/language/guide/DifferentialEquations.html. Accessed 24 Apr. 2026.

Hartman, Philip. Ordinary Differential Equations. 2nd ed., Birkhauser, 1982.

Hirsch, Morris W., et al. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed., Academic Press, 2013.

“Introduction to Differential Calculus.” GeeksforGeeks, 31 Jan. 2026, www.geeksforgeeks.org/differential-calculus/. Accessed 24 Apr. 2026.

Kocak, Hueseyin. Differential and Difference Equations through Computer Experiments. Springer-Verlag, 1986.

Krantz, Steven G., and George F. Simmons. Differential Equations: Theory, Technique, and Practice. 3rd ed., Chapman & Hall/CRC, 2022.

Noonburg, V. W. Differential Equations: From Calculus to Dynamical Systems. 2nd ed., MAA Press, an imprint of the American Mathematical Society, 2019.

Full Article

  • Type of physical science: Mathematical methods
  • Field of study: Calculus

Many laws of physics are best expressed by prescribing relationships between a function describing the phenomenon and its rate of change. Once these laws are supplemented by other conditions, such as the value of the functions at a specified time, it becomes possible to find out what actually happens.

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, as described by Newton’s law of cooling.

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 spaceship and ask where it will be one year later. This is called the “initial value problem.”

One important advantage of the differential equation’s 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 usually 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 the tool of choice. Perhaps more importantly, 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 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 and other fields such as epidemiology and finance.

In application to mechanics, the success has been spectacular. Almost all features of the solar system have been accounted for (notable examples are the rings of Saturn, the tidal locking that ensures the Moon always faces the Earth, and the spin-orbit resonance that causes Mercury to rotate 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 importantly, 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 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 in 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 in 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 a 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 receivers, televisions, 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 that 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 optimal 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 continued to develop through the early twentieth century, although it gained broader prominence later with the rise of dynamical systems theory. In both the West and the East, it was used mainly by mechanical engineers and those who studied celestial mechanics, but in the East, it was also used 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 J. Moser in the West and A. N. Kolmogorov, V. I. Arnold, and Y. Sinai in the East.

One important tool that made progress quicker and easier was the increasing availability of computers. The graphics made it easy to obtain a geometric intuition of phenomena and to test conjectures. Computers greatly enhanced the ways of obtaining quantitative predictions that were useful in applied problems, complementing earlier analytical and approximate methods.

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 Feigenbaum that renormalization group methods could produce quantitative results for chaotic systems. From that time on, the field experienced explosive growth, and one can 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 find important results in many journals devoted to chemistry, physics, or biology.

Principal terms

DIFFERENTIAL EQUATION: a relationship between the derivatives of one or more functions and the functions themselves

GENERAL SOLUTION OF A DIFFERENTIAL EQUATION: a formula involving arbitrary constants such that, by assigning values to the constants, one gets all the solutions of the differential equation

ORDER OF THE DIFFERENTIAL EQUATION: the order of the derivative of the highest order appearing in the differential equation

PARAMETERS: variables that do not enter the differential equation but that correspond to magnitudes that can be set from the exterior

SOLUTION OF A DIFFERENTIAL EQUATION: a function or group of functions whose derivatives are related to the functions in the way prescribed by the differential equation


Bibliography

Alazard, Thomas. Analysis and Partial Differential Equations. Springer, 2024.

Arnold, V. I. Ordinary Differential Equations. Translated by Richard A. Silverman, MIT Press, 1973.

Boyce, William E., and Richard C. DiPrima. Elementary Differential Equations and Boundary Value Problems. 4th ed., Wiley, 1986.

Braun, Martin. Differential Equations and Their Applications. 2nd ed., Springer-Verlag, 1978.

“Differential Equations.” Math Is Fun, www.mathsisfun.com/calculus/differential-equations.html. Accessed 24 Apr. 2026.

“Differential Equations.” Paul’s Online Notes, 26 June 2023, tutorial.math.lamar.edu/classes/de/de.aspx. Accessed 24 Apr. 2026.

“Differential Equations.” Wolfram, reference.wolfram.com/language/guide/DifferentialEquations.html. Accessed 24 Apr. 2026.

Hartman, Philip. Ordinary Differential Equations. 2nd ed., Birkhauser, 1982.

Hirsch, Morris W., et al. Differential Equations, Dynamical Systems, and an Introduction to Chaos. 3rd ed., Academic Press, 2013.

“Introduction to Differential Calculus.” GeeksforGeeks, 31 Jan. 2026, www.geeksforgeeks.org/differential-calculus/. Accessed 24 Apr. 2026.

Kocak, Hueseyin. Differential and Difference Equations through Computer Experiments. Springer-Verlag, 1986.

Krantz, Steven G., and George F. Simmons. Differential Equations: Theory, Technique, and Practice. 3rd ed., Chapman & Hall/CRC, 2022.

Noonburg, V. W. Differential Equations: From Calculus to Dynamical Systems. 2nd ed., MAA Press, an imprint of the American Mathematical Society, 2019.

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