A
differential equation is a
mathematicalMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
equationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
for an unknown
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of one or several
variablesIn mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
that relates the values of the function itself and its
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of various orders. Differential equations play a prominent role in
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
,
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a
deterministicIn mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.Examples:...
relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in
classical mechanicsIn physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, where the motion of a body is described by its position and velocity as the time value varies.
Newton's lawsNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.
Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions —the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a selfcontained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
Directions of study
The study of differential equations is a wide field in
pureBroadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
and
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
,
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
meteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
, and
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve reallife problems may not necessarily be directly solvable, i.e. do not have
closed formIn mathematics, an expression is said to be a closedform expression if it can be expressed analytically in terms of a bounded number of certain "wellknown" functions...
solutions. Instead, solutions can be approximated using
numerical methodsNumerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
.
Mathematicians also study
weak solutionIn mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
s (relying on
weak derivativeIn mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition. Definition :Let u be a function in the...
s), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type.
The study of the stability of solutions of differential equations is known as
stability theoryIn mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...
.
Nomenclature
The theory of differential equations is quite developed and the methods
used to study them vary significantly with the type of the equation.
 An ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
(ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vectorvaluedA vectorvalued function also referred to as a vector function is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinitedimensional vectors. The input of a vectorvalued function could be a scalar or a vector...
or matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
valued: this corresponds to considering a system of ordinary differential equations for a single function.
 Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are firstorder and secondorder differential equations. For example, Bessel's differential equation
 is a secondorder differential equation. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.
 A partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for secondorder linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.
Both ordinary and partial differential equations are broadly classified as
linear and
nonlinear. A differential equation is
linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and
nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations.
Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a
constant coefficient linear differential equation.
There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of
chaosChaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and wellposedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).
Linear differential equations frequently appear as
approximationsIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).
Classification summary
The mathematical definitions for the various classifications of differential equation can be collected as follows.
Ordinary DE classification
See main article: Ordinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
.
In the table below,
All differential equations are of
order n and arbitrary
degree d.
F is an
implicit functionThe implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
of: an independent variable
x, a dependent variable
y (a function of
x), and integer derivatives of
y (
fractional derivativesFractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
are in fact possible, but not considered here).
y may in general be a vector valued function:

so x is an element of the vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
R, y an element of a vector space of dimension m, , where R is the set of real numbers, is the cartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of R with itself m times to form an mtuple of real numbers.
This leads to a system of differential equations to be solved for y_{1}, y_{2},...y_{m}.
y is characterized by the function mappingIn most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
.
r(x) is called a source term in x, and A(x) is an arbitrary function, both assumed continuous in x on defined intervals.
Notice the mapping from or corresponds to the map from x, y, and the n or (n1) derivatives of y to the solution, in general implicit.
Examples
In the first group of examples, let u be an unknown function of x, and c and ω are known constants.
 Inhomogeneous firstorder linear constant coefficient ordinary differential equation:


 Homogeneous secondorder linear ordinary differential equation:


 Homogeneous secondorder linear constant coefficient ordinary differential equation describing the harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = k \vec x \, where k is a positive constant....
:


 Firstorder nonlinear ordinary differential equation:


 Secondorder nonlinear ordinary differential equation describing the motion of a pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
of length L:


In the next group of examples, the unknown function u depends on two variables x and t or x and y.
 Homogeneous firstorder linear partial differential equation:


 Homogeneous secondorder linear constant coefficient partial differential equation of elliptic type, the Laplace equation:


 Thirdorder nonlinear partial differential equation, the Korteweg–de Vries equation
In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a nonlinear partial differential equation whose solutions can be exactly and precisely specified...
:


Related concepts
 A delay differential equation
In mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
(DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
 A stochastic differential equation
A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
(SDE) is an equation in which the unknown quantity is a stochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
and the equation involves some known stochastic processes, for example, the Wiener processIn mathematics, the Wiener process is a continuoustime stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
in the case of diffusion equations.
Connection to difference equations
The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.
Universality of mathematical description
Many fundamental laws of physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
can be formulated as differential equations. In biologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
and economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same secondorder partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
, the wave equationThe wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph FourierJean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
, is governed by another secondorder partial differential equation, the heat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
. It turned out that many diffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
processes, while seemingly different, are described by the same equation; Black–Scholes equation in finance is for instance, related to the heat equation.
Exact solutions
Some differential equations have solutions which can be written in an exact and closed form. Several important classes are given here.
In the table below, H(x), Z(x), H(y), Z(y), or H(x,y), Z(x,y) are any integrable functions of x or y (or both), and A, B, C, I, L, N, M are all constants. In general A, B, C, I, L, are real numbers, but N, M, P and Q may be complex. The differential equations are in their equivalent and alternative forms which lead to the solution through integrationIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
.
\frac{x}{2} \right )} + Q\cos{\left ( \sqrt{\left  I^24L \right }\frac{x}{2} \right )} \right ] \,\!

 9  
where are the d solutions of the polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
of degreeThe degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
d:
}
Note that 3 and 4 are special cases of 7, they are relatively common cases and included for completeness.
Similarly 8 is a special case of 9, but 8 is a relatively common form, particularly in simple physical and engineering problems.
Biology
 Verhulst equation – biological population growth
 von Bertalanffy model – biological individual growth
 Lotka–Volterra equations – biological population dynamics
 Replicator dynamics – may be found in theoretical biology
 HodgkinHuxley model
The Hodgkin–Huxley model is a mathematical model that describes how action potentials in neurons are initiated and propagated....
 neural action potentials
Economics
 The Black–Scholes PDE
 Exogenous growth model
The neoclassical growth model, also known as the Solow–Swan growth model or exogenous growth model, is a class of economic models of longrun economic growth set within the framework of neoclassical economics...
 Malthusian growth model
The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest...
 The VidaleWolfe advertising model
The Sethi model was developed by Suresh P. Sethi and describes the process of how sales evolve over time in response to advertising. The rate of change in sales depend on three effects: response to advertising that acts positively on the unsold portion of the market, the loss due to forgetting or...
See also
 Complex differential equation
A complex differential equation is a differential equation whose solutions are functions of a complex variable.Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied...
 Exact differential equation
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.Definition:...
 Integral equations
 Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
 Picard–Lindelöf theorem on existence and uniqueness of solutions
 Numerical methods
External links