In mathematics,

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

is a fundamental concept that is used in many scientific areas. Many of the differential equations that are used have received specific names, which are listed in this article.

Pure mathematics

Calabi flow In the mathematical fields of differential geometry and geometric analysis, the Calabi flow is a geometric flow which deforms a Kähler metric on a complex manifold. Precisely, given a Kähler manifold , the Calabi flow is given by: :\frac=\frac, wh ...

in the study of Calabi-Yau manifolds * Cauchy–Riemann equations in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

Clairaut's equation In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form :y(x)=x\frac+f\left(\frac\right) where ''f'' is continuously differentiable. It is a particular case of the Lagrange differential e ...

* Hypergeometric differential equation *Equations for a minimal surface in

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analo ...

, used to prove the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...

of orthogonal polynomials and separable partial differential equations * Universal differential equation

Physics

Classical mechanics

So long as the force acting on a particle is known,

Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

, which is called the ''equation of motion''. Classical mechanics for particles finds its generalization in

continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...

Electrodynamics

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...

are a set of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s that, together with the

Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...

law, form the foundation of

classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...

, classical

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...

, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and

magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

s are generated and altered by each other and by

charges Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...

and currents. They are named after the Scottish physicist and mathematician

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

, who published an early form of those equations between 1861 and 1862.

Poynting's theorem In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting. It states that in a given volume, the stored energy changes at a rate given by the work ...

is a statement of the

conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...

for

electromagnetic fields An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...

, expressed as a partial differential equation.

General relativity

The Einstein field equations (EFE; also known as "Einstein's equations") are a set of ten

s in

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

's general theory of relativity which describe the fundamental interaction of

gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

as a result of

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

being

curved In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ap ...

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...

and

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...

. First published by Einstein in 1915 as a

tensor equation In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

, the EFE equate local spacetime

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

(expressed by the

Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field eq ...

) with the local energy and

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

within that spacetime (expressed by the

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...

). *

Friedmann equations The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedman ...

Geodesic equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection ...

in general relativity *

Mathisson–Papapetrou–Dixon equations In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field. Other equations with similar names and mathematical forms are the Mathisson� ...

Thermodynamics and statistical mechanics

Diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...

Heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...

* Kardar-Parisi-Zhang equation *

Maxwell relations file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...

Quantum mechanics and quantum field theory

* Dirac equation, the relativistic wave equation for electrons and positrons * Klein–Gordon equation * Schrödinger's equation *

Schwinger–Dyson equation The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between correlation functions in quantum field theories (QFTs). They are also referred to as the Euler– ...

* Yang-Mills equations in

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

Astrophysics

* Lane-Emden equation * Emden–Chandrasekhar equation

Engineering

Fluid dynamics and hydrology

Biology and medicine

Predator–prey equations

The

Lotka–Volterra equations The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a pred ...

, also known as the predator–prey equations, are a pair of first-order,

non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

, differential equations frequently used to describe the

population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...

of two species that interact, one as a predator and the other as prey. Another set of equations describing predator-prey dynamics are the

Arditi–Ginzburg equations The Arditi–Ginzburg equations describes ratio dependent predator–prey dynamics. Where ''N'' is the population of a prey species and ''P'' that of a predator, the population dynamics are described by the following two equations: \begin \frac & = ...

, where ratio dependence is a dominant feature.

Chemistry

The ''rate law'' or

rate equation In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reacti ...

for a

chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...

is a differential equation that links the

reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...

with concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial

reaction order In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial react ...

s). To determine the rate equation for a particular system one combines the reaction rate with a

mass balance In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have bee ...

for the system.Kenneth A. Connors ''Chemical Kinetics, the study of reaction rates in solution'', 1991, VCH Publishers. In addition, a range of differential equations are present in the study of

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

and

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

. The Allen-Cahn and Cahn–Hilliard equations have been applied to study

phase separation Phase separation is the creation of two distinct phases from a single homogeneous mixture. The most common type of phase separation is between two immiscible liquids, such as oil and water. Colloids are formed by phase separation, though n ...

. * Master equation *

Streeter–Phelps equation The Streeter–Phelps equation is used in the study of water pollution as a water quality modelling tool. The model describes how dissolved oxygen (DO) decreases in a river or stream along a certain distance by degradation of biochemical oxygen ...

water quality modeling Water quality modeling involves water quality based data using mathematical simulation techniques. Water quality modeling helps people understand the eminence of water quality issues and models provide evidence for policy makers to make decisions ...

Economics and finance

References

{{Differential equations topics Differential equations