NationMaster - Encyclopedia: Ordinary differential equation

In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In an experimental design, the independent variable (argument of a function, also called a predictor variable) is the variable that is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... This article is about derivatives and differentiation in mathematical calculus. ...

A simple example is Newton's second law of motion, which leads to the differential equation Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

$m frac{d^2 x(t)}{dt^2} = F(x(t)),,$

for the motion of a particle of mass m. In general, the force F depends upon the position of the particle x(t) at time t, and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).

Ordinary differential equations are to be distinguished from partial differential equations where there are several independent variables involving partial derivatives. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... 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). ...

Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler. Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... Bernoulli family tree The Bernoullis were a family of traders and scholars from Basel, Switzerland. ... Jacopo Francesco Riccati (28 May 1676 - 15 April 1754) was an Italian mathematician, from Venice. ... Alexis Claude Clairault (or Clairaut) (May 3, 1713 - May 17, 1765) was a French mathematician. ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean Le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations). In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ...

The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law.

1 Definitions
- 1.1 Ordinary differential equation
- 1.2 Solutions
2 Examples
3 Reduction to a first order system
4 Linear ordinary differential equations
5 Theories of ODEs
6 See also
7 Bibliography
8 External links

Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... A projectile is any object sent through space by the application of a force. ... For other uses, see Cannon (disambiguation). ...

Definitions

Ordinary differential equation

Let y be an unknown function

$y: mathbb{R} to mathbb{R}$

in x with $y (i)$ the i-th derivative of y, then a function This article is about derivatives and differentiation in mathematical calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

$F(x,y,y', dots, y^{(n-1)})=y^{(n)}$

is called an ordinary differential equation (ODE) of order n. For vector valued functions A graph of the vector-valued function <2Cos(t),4Sin(t),t> A vector-valued function is a mathematical function that maps real numbers onto vectors. ...

$y: mathbb{R} to mathbb{R}^m$

F is called a system of ordinary differential equations of dimension m.

When a differential equation of order n has the form

$Fleft(x, y, y', y'', dots, y^{(n)}right) = 0$

it is called an implicit differential equation whereas the form

$Fleft(x, y, y', y'', dots, y^{(n-1)}right) = y^{(n)}$

is called an explicit differential equation.

A differential equation not depending on x is called autonomous. In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not depend on the independent variable. ...

A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

$y^{(n)} = sum_{i=1}^{n-1} a_i(x) y^{(i)} + r(x)$

with a_i(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear differential equation is called homogeneous, otherwise it is called non-homogeneous or inhomogeneous.

Solutions

Given a differential equation

$F(x,y,y', dots, y^{(n)})=0$

a function

$u: I subset mathbb{R} to mathbb{R}$

is called solution or integral curve for F, if u is n-times differentiable on I, F is defined for all

$(x,u,u', dots, u^{(n)}) quad x in I$

and

$F(x,u,u', dots, u^{(n)})=0 quad x in I.$

Given two solutions

$u: J subset mathbb{R} to mathbb{R}$

and

$v: I subset mathbb{R} to mathbb{R}$

u is called an extension of v if I ⊂ J and

$u(x) = v(x) quad x in I.,$

A solution which has no extension is called a global solution.

A general solution of an n-th order equation is a solution containing $n$ arbitrary variables, corresponding to n 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 or Boundary Conditions'. A singular solution is a solution that can't be derived from the general solution. In calculus, the indefinite integral of a given function (i. ... A singular solution of a differential equation is a solution that satisfies the following conditions: It solves the original differential equation. ...

Examples

Main article: Examples of differential equations

// A separable first order linear ordinary differential equation A separable linear ordinary differential equation of the first order has the general form: where f(t) is some known function. ...

Reduction to a first order system

Any differential equation of order n can be written as a system of n first-order differential equations. Given an explicit ordinary differential equation of order n and dimension 1,

$Fleft(x, y, y', y'', dots, y^{(n-1)}right) = y^{(n)}$

we define a new family of unknown functions

$y_n := y^{(n-1)}.!$

We can then rewrite the original differential equation as a system of differential equations with order 1 and dimension n.

$y_1^' = y_2$

$vdots$

$y_n^' = F(y_n, dots, y_1, x).$

which can be written concisely in vector notation as

$mathbf{y}^'=mathbf{F}(mathbf{y}, x)$

with

$mathbf{y}:=(y,ldots,y_n).$

Linear ordinary differential equations

Main article: Linear differential equation

A well understood particular class of differential equations are linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1 In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

$y_i'(x) = sum_{j=1}^n a_{i,j}(x) y_j + b_i(x) , mathrm{,} quad i = 1,ldots,n$

which we can write concisely using vector notation as

$mathbf{y}^'(x) = mathbf{A}(x) mathbf{y}(x) + mathbf{b}(x)$

with

$mathbf{y}(x):=(y_1(x),ldots,y_n(x))$

$mathbf{b}(x):=(b_1(x),ldots,b_n(x))$

$mathbf{A}(x):=(a_{i,j}(x)) , mathrm{,} quad i,j = 1,ldots,n.$

Homogeneous equations

The set of solutions for a system of homogeneous linear differential equations of order 1 and dimension n

$mathbf{y}^'(x) = mathbf{A}(x) mathbf{y}(x)$

forms an n-dimensional vector space. Given a basis for this vector space $mathbf{z}_1(x), ldots, mathbf{z}_n(x)$ , which is called a fundamental system, every solution $mathbf{s}(x)$ can be written as In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

$mathbf{s}(x) = sum_{i=1}^{n} c_i mathbf{z}_i(x).$

The n × n matrix

$mathbf{Z}(x) := (mathbf{z}_1(x), ldots, mathbf{z}_n(x))$

is called fundamental matrix. In general there is no method to explicitly construct a fundamental system, but if one solution is known d'Alembert reduction can be used to reduce the dimension of the differential equation by one.

Non-homogeneous equations

The set of solutions for a system of inhomogeneous linear differential equations of order 1 and dimension n

$mathbf{y}^'(x) = mathbf{A}(x) mathbf{y}(x) + mathbf{b}(x)$

can be constructed by finding the fundamental system $mathbf{z}_1(x), ldots, mathbf{z}_n(x)$ to the corresponding homogeneous equation and one particular solution $mathbf{p}(x)$ to the inhomogeneous equation. Every solution $mathbf{s}(x)$ to inhomogeneous equation can then be written as

$mathbf{s}(x) = sum_{i=1}^{n} c_i mathbf{z}_i(x) + mathbf{p}(x).$

A particular solution to the inhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters. In mathematics, the method of undetermined coefficients is an approach to solving certain ordinary differential equations and recurrence relations. ... In mathematics, variation of parameters or variation of constants is a method used to solve inhomogeneous linear ordinary differential equations. ...

Fundamental systems for homogeneous equations with constant coefficients

For a system of homogeneous linear differential equations with constant coefficients

$mathbf{y}^'(x) = mathbf{A} mathbf{y}(x)$

we can explicitly construct a fundamental system. The system can be written as a matrix differential equation

$mathbf{Y}^' = mathbf{A} mathbf{Y}$

with solution as a matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. ...

$e^{x mathbf{A}}$

which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal form In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M...

$e^{x mathbf{A}} = e^{x mathbf{C}^{-1} mathbf{J} mathbf{C}^{1}} = mathbf{C}^{-1} e^{x mathbf{J}} mathbf{C}^{1}$

and then evaluate the Jordan blocks In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard...

$J_i = begin{bmatrix} lambda_i & 1 & ; & ; ; & ddots & ddots & ; ; & ; & ddots & 1 ; & ; & ; & lambda_i end{bmatrix}$

of J separately as

$e^{x mathbf{J_i}} = e^{lambda_i x} begin{bmatrix} 1 & x & frac{x^2}{2} & dots & frac{x^{n-1}}{(n-1)!} ; & ddots & ddots & ddots & vdots ; & ; & ddots & ddots & frac{x^2}{2} ; & ; & ; & ddots & x ; & ; & ; & ; & 1 end{bmatrix} .$

Theories of ODEs

Singular solutions

The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was 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. A singular solution of a differential equation is a solution that satisfies the following conditions: It solves the original differential equation. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Felice Casorati (December 17, 1835 â€“ September 11, 1890) was an Italian mathematician best known for the Weierstrassâ€“Casorati theorem in complex analysis. ...

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 the differential equation meets its limitations very soon unless complex numbers are introduced. 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 to be, not 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 of this function. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ... Johann Carl Friedrich Gauss (pronounced , ; in German usually GauÃŸ, Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Fuchsian theory

Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations. Immanuel Lazarus Fuchs (5 May 1833 - 26 April 1902) was a German mathematician. ... A picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 â€“ August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. ...

Lie's theory

From 1870 Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact (Berührungstransformationen). Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... To meet Wikipedias quality standards and make it more accessible, this article may require cleanup. ... This page is a candidate for speedy deletion, because: No meaningful content If you disagree with its speedy deletion, please explain why on its talk page or at Wikipedia:Speedy deletions. ...

Sturm-Liouville theory

Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients. In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles FranÃ§ois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

Bibliography

A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
Hartman, Philip, Ordinary Differential Equations, 2nd Ed., Society for Industrial & Applied Math, 2002. ISBN 0-89871-510-5.
W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
E.L. Ince, Ordinary Differential Equations, Dover Publications, 1958, ISBN 0486603490
Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0-486-49510-8

Witold Hurewicz (June 29, 1904 - September 6, 1956) was a Polish mathematician. ...

External links

Wikibooks has a book on the topic of

Calculus/Ordinary differential equations

Differential Equations at the Open Directory Project (includes a list of software for solving differential equations).
EqWorld: The World of Mathematical Equations, containing a list of ordinary differential equations with their solutions.
Online Notes / Differential Equations by Paul Dawkins, Lamar University.
Differential Equations, S.O.S. Mathematics.
A primer on analytical solution of differential equations from the Holistic Numerical Methods Institute, University of South Florida.
A concise introductory course in ordinary differential equations

Categories: Differential calculus | Ordinary differential equations

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... The Open Directory Project (ODP), also known as dmoz (from , its original domain name), is a multilingual open content directory of World Wide Web links owned by Netscape that is constructed and maintained by a community of volunteer editors. ... Lamar University is a four-year university located in Beaumont, Texas, USA, and a member of the Texas State University System. ... S.O.S. Mathematics is a website which provides students with math-related materials which are intended to refresh or reinforce what they already know about mathematics. ...

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