In mathematics, a differential equation is an

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

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, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on

qualitative Qualitative descriptions or distinctions are based on some quality or characteristic rather than on some quantity or measured value. Qualitative may also refer to: *Qualitative property, a property that can be observed but not measured numericall ...

analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

History

Differential equations first came into existence with the invention of calculus by

Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...

and Leibniz. In Chapter 2 of his 1671 work ''Methodus fluxionum et Serierum Infinitarum'', Isaac Newton listed three kinds of differential equations: :

x_1 \frac &+ x_2 \frac = y \end

In all these cases, is an unknown function of (or of and ), and is a given function. He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form :

y'+ P(x)y = Q(x)y^n\,

for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a vibrating string such as that of a

musical instrument A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person who pl ...

was studied by Jean le Rond d'Alembert, Leonhard Euler,

Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...

, and

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiawave equation, and within ten years Euler discovered the three-dimensional wave equation.Speiser, David.
Discovering the Principles of Mechanics 1600-1800
', p. 191 (Basel: Birkhäuser, 2008). The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in ''Théorie analytique de la chaleur'' (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his

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 ...

for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.

Example

In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) 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 modeling 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 a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

Types

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

An ordinary differential equation (''ODE'') is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a variable (often denoted ), which, therefore, ''depends'' on . Thus is often called the

independent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...

of the equation. The term "''ordinary''" is used in contrast with the term

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 ...

, which may be with respect to ''more than'' one independent variable. Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals. Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function). As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

Partial differential equations

(''PDE'') is a differential equation that contains unknown

multivariable function In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function ...

s and their partial derivatives. (This is in contrast to

ordinary differential equations 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 deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat,

electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...

, electrodynamics,

fluid flow In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the cl ...

, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

, partial differential equations often model multidimensional systems.

Stochastic partial differential equations Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have ...

generalize partial differential equations for modeling randomness.

Non-linear differential equations

A non-linear differential equation is a differential equation that is not a

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...

in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness 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 The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the N ...

). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution. Linear differential equations frequently appear as approximations 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).

Equation order

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a ''first-order differential equation'', an equation containing the second derivative is a ''second-order differential equation'', and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

Examples

In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between '' linear'' and ''nonlinear'' differential equations, and between ''homogeneous'' differential equations and ''heterogeneous'' ones. * Heterogeneous first-order linear constant coefficient ordinary differential equation: *:

\frac = cu+x^2.

* Homogeneous second-order linear ordinary differential equation: *:

\frac - x\frac + u = 0.

* Homogeneous second-order linear constant coefficient ordinary differential equation describing the

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

: *:

\frac + \omega^2u = 0.

* Heterogeneous first-order nonlinear ordinary differential equation: *:

\frac = u^2 + 4.

* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length ''L'': *:

L\frac + g\sin u = 0.

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''. * Homogeneous first-order linear partial differential equation: *:

\frac + t\frac = 0.

* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation: *:

\frac + \frac = 0.

* Homogeneous third-order non-linear partial differential equation: *:

\frac = 6u\frac - \frac.

Existence of solutions

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest. For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point

(a, b)

in the xy-plane, define some rectangular region

Z

, such that

Z =, m times

, p The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

/math> and

(a, b)

is in the interior of

Z

. If we are given a differential equation

\frac = g(x, y)

and the condition that

y = b

when

x = a

, then there is locally a solution to this problem if

g(x, y)

and

\frac

are both continuous on

Z

. This solution exists on some interval with its center at

a

. The solution may not be unique. (See Ordinary differential equation for other results.) However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order: :

f_(x)\frac + \cdots + f_(x)\frac + f_(x)y = g(x)

such that :

\begin
    y(x_) &= y_, &
   y'(x_) &= y'_, &
  y''(x_) &= y''_, &
      \ldots
\end

For any nonzero

f_(x)

, if

\

and

g

are continuous on some interval containing

x_

y

is unique and exists.

Related concepts

* A delay differential equation (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. * An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation. * A stochastic differential equation (SDE) is an equation in which the unknown quantity is a

stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations. *A

stochastic partial differential equation Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have ...

(SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

and

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

. * An ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use

pseudo-differential operators In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in m ...

instead of

differential operators In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

. * A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

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 the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Applications

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. 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 modeling 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 real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods. Many fundamental laws of physics and

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

can be formulated as differential equations. In biology and economics, 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 the 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 second-order

, the wave equation, 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 Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...

, is governed by another second-order partial differential equation, the

. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation. The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See

List of named differential equations In mathematics, differential equation 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 flo ...

Software

Some

CAS Cas may refer to: * Caș, a type of cheese made in Romania * ' (1886–) Czech magazine associated with Tomáš Garrigue Masaryk * '' Čas'' (19 April 1945–February 1948), the official, daily newspaper of the Democratic Party of Slovakia * ''CA ...

software can solve differential equations. These

software and their commands are worth mentioning: * Maple: dsolve *

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...

: DSolve[] * Maxima: ode2(equation, y, x) * SageMath: desolve() * SymPy: sympy.solvers.ode.dsolve(equation) * Xcas: desolve(y'=k*y,y)

References

External links

*
Lectures on Differential Equations
MIT Open CourseWare Videos
Online Notes / Differential Equations
Paul Dawkins, Lamar University
Differential Equations
S.O.S. Mathematics

Introduction to modeling by means of differential equations, with critical remarks.
Mathematical Assistant on Web
Symbolic ODE tool, using Maxima
Exact Solutions of Ordinary Differential Equations
MATLAB models
Notes on Diffy Qs: Differential Equations for Engineers
An introductory textbook on differential equations by Jiri Lebl of UIUC
Khan Academy Video playlist on differential equations
Topics covered in a first year course in differential equations.
MathDiscuss Video playlist on differential equations
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