In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ordinary differential equation (ODE) is a
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, ...
whose unknown(s) consists of one (or more) function(s) of one
variable and involves the
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 of those functions.
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 function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
which may be with respect to ''more than'' one independent variable.
Differential equations
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
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 F ...
of the form
:
where , ..., and are arbitrary
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s that do not need to be linear, and are the successive derivatives of the unknown function of the variable .
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most
elementary and
special functions 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 ...
and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
are solutions of linear differential equations (see
Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example
Riccati equation).
Some ODEs can be solved explicitly in terms of known functions and
integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
. When that is not possible, the equation for computing the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the solutions may be useful. For applied problems,
numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as " numerical integration", although this term can also ...
can supply an approximation of the solution.
Background
Ordinary differential equations (ODEs) arise in many contexts of mathematics and
social
Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not.
Etymology
The word "social" derives from ...
and
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
analytical mechanics. Scientific fields include much of
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 ...
and
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
(celestial mechanics),
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
(weather modeling),
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
(reaction rates),
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 ...
(infectious diseases, genetic variation),
ecology
Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
and
population modeling (population competition),
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 anal ...
(stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including
Newton,
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, the
Bernoulli family,
Riccati,
Clairaut,
d'Alembert, and
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
.
A simple example is
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 mo ...
of motion — the relationship between the displacement ''x'' and the time ''t'' of an object under the force ''F'', is given by the differential equation
:
which constrains the
motion of a particle of constant mass ''m''. In general, ''F'' is a function of the position ''x''(''t'') of the particle at time ''t''. The unknown function ''x''(''t'') appears on both sides of the differential equation, and is indicated in the notation ''F''(''x''(''t'')).
Definitions
In what follows, let ''y'' be a
dependent 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 dema ...
and ''x'' an
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 dema ...
, and ''y'' = ''f''(''x'') is an unknown function of ''x''. The
notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the
Leibniz's notation is more useful for differentiation and
integration, whereas
Lagrange's notation is more useful for representing derivatives of any order compactly, and
Newton's notation is often used in physics for representing derivatives of low order with respect to time.
General definition
Given ''F'', a function of ''x'', ''y'', and derivatives of ''y''. Then an equation of the form
:
is called an ''
explicit
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge
Explicit knowledge (also expressive knowledge) is knowledge that can be readily articulated, codified, stored and accessed. It can be expres ...
ordinary differential equation of order n''.
More generally, an ''
implicit
Implicit may refer to:
Mathematics
* Implicit function
* Implicit function theorem
* Implicit curve
* Implicit surface
* Implicit differential equation
Other uses
* Implicit assumption, in logic
* Implicit-association test, in social psycholog ...
'' ordinary differential equation of order ''n'' takes the form:
:
There are further classifications:
System of ODEs
A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(''x'') =
1(''x''), ''y''2(''x''),..., ''ym''(''x'')">'y''1(''x''), ''y''2(''x''),..., ''ym''(''x'') and F is a
vector-valued function of y and its derivatives, then
:
is an ''explicit system of ordinary differential equations'' of ''order'' ''n'' and ''dimension'' ''m''. In
column vector form:
:
These are not necessarily linear. The ''implicit'' analogue is:
:
where 0 = (0, 0, ..., 0) is the
zero vector. In matrix form
:
For a system of the form
, some sources also require that the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
be
non-singular in order to call this an implicit ODE
ystem an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed
differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems.
Presumably for additional derivatives, the
Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that
any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.
The behavior of a system of ODEs can be visualized through the use of a
phase portrait.
Solutions
Given a differential equation
:
a function , where ''I'' is an interval, is called a ''solution'' or
integral curve for ''F'', if ''u'' is ''n''-times differentiable on ''I'', and
:
Given two solutions and , ''u'' is called an ''extension'' of ''v'' if and
:
A solution that has no extension is called a ''maximal solution''. A solution defined on all of R is called a ''global solution''.
A ''general solution'' of an ''n''th-order equation is a solution containing ''n'' arbitrary independent
constants of integration. A ''particular solution'' is derived from the general solution by setting the constants to particular values, often chosen to fulfill set '
initial conditions or
boundary conditions'. A
singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
In the context of linear ODE, the terminology ''particular solution'' can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the ''homogeneous'' solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the
guessing method section in this article, and is frequently used when discussing the
method of undetermined coefficients and
variation of parameters.
Solutions of Finite Duration
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because they will being non-Lipschitz functions at their ending time, they don´t stand uniqueness of solutions of Lipschitz differential equations.
As example, the equation:
:
Admits the finite duration solution:
:
Theories
Singular solutions
The theory of
singular solutions of ordinary and
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century has it received special attention. A valuable but little-known work on the subject is that of Houtain (1854).
Darboux (from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably
Casorati and
Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
Reduction to quadratures
The primitive attempt in dealing with differential equations had in view a reduction to
quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the ''n''th degree, so it was the hope of analysts to find a general method for integrating any differential equation.
Gauss (1799) showed, however, that complex differential equations require
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. Hence, analysts began to substitute the study of functions, thus opening a new and fertile field.
Cauchy was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and, if so, what are the characteristic properties.
Fuchsian theory
Two memoirs by
Fuchs inspired a novel approach, subsequently elaborated by Thomé and
Frobenius. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868.
Clebsch (1873) attacked the theory along lines parallel to those in his theory of
Abelian integrals. As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces ''f'' = 0 under rational one-to-one transformations.
Lie's theory
From 1870,
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius S ...
's work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, be referred to a common source, and that ordinary differential equations that admit the same
infinitesimal transformations present comparable integration difficulties. He also emphasized the subject of
transformations of contact.
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations.
A general solution approach uses the symmetry property of differential equations, the continuous
infinitesimal transformations of solutions to solutions (
Lie theory). Continuous
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
,
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
, and
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 mult ...
are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its
Lax pairs, recursion operators,
Bäcklund transform, and finally finding exact analytic solutions to DE.
Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
Sturm–Liouville theory
Sturm–Liouville theory is a theory of a special type of second order linear ordinary differential equation. Their solutions are based on
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
and corresponding
eigenfunctions of linear operators defined via second-order
homogeneous linear equations. The problems are identified as Sturm-Liouville Problems (SLP) and are named after
J.C.F. Sturm and
J. Liouville, who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations.
Existence and uniqueness of solutions
There are several theorems that establish existence and uniqueness of solutions to
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...
s involving ODEs both locally and globally. The two main theorems are
:
In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of
Grönwall's inequality are met.
Also, uniqueness theorems like the Lipschitz one above do not apply to
DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone.
Local existence and uniqueness theorem simplified
The theorem can be stated simply as follows.
[Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ] For the equation and initial value problem:
if ''F'' and ∂''F''/∂''y'' are continuous in a closed rectangle