In mathematics, a differential equation is an
equation that relates one or more unknown
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s and their
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s.
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, and
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
.
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 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 and
Leibniz. In Chapter 2 of his 1671 work
''Methodus fluxionum et Serierum Infinitarum'', Isaac Newton listed three kinds of differential equations:
:
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
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
proposed the
Bernoulli differential equation
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form
: y'+ P(x)y = Q(x)y^n,
where n is a real number. Some authors allow any real n, whereas others require that n not be 0 or 1. The ...
in 1695. This is an
ordinary differential equation of the form
:
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 was studied by
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
,
Leonhard Euler,
Daniel Bernoulli, and
Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional
wave 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
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
was developed in the 1750s by Euler and Lagrange in connection with their studies of the
tautochrone
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ...
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 physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
.
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
In the study of heat transfer, Newton's law of cooling is a physical law which states that
The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.
The law is frequently q ...
, 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 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 of the equation. The term "''ordinary''" is used in contrast with the term
partial differential equation, which may be with respect to ''more than'' one independent variable.
Linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
s 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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
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
A
partial differential equation (''PDE'') is a differential equation that contains unknown
multivariable functions and their
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
. (This is in contrast to
ordinary differential equations, 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
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...
.
PDEs can be used to describe a wide variety of phenomena in nature such as
sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' b ...
,
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
,
electrostatics,
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
fluid flow,
elasticity, or
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 chemistr ...
. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional
dynamical systems, partial differential equations often model
multidimensional systems.
Stochastic partial differential equations generalize partial differential equations for modeling
randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
.
Non-linear differential equations
A non-linear differential equation is a differential equation that is not a
linear equation 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
Chaos or CHAOS may refer to:
Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
. 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). 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
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
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:
*:
* Homogeneous second-order linear ordinary differential equation:
*:
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the
harmonic oscillator:
*:
* Heterogeneous first-order nonlinear ordinary differential equation:
*:
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a
pendulum 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 first-order linear partial differential equation:
*:
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the
Laplace equation:
*:
* Homogeneous third-order non-linear partial differential equation:
*:
Existence of solutions
Solving differential equations is not like solving
algebraic equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
. 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
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees t ...
gives one set of circumstances in which a solution exists. Given any point
in the xy-plane, define some rectangular region
, such that