In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects. They are used to distinguish between two types of quantities being considered, separating them into those available at the start of a, and one or more of its derivatives In calculus 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 vehicle with respect to time is the vehicle's instantaneous velocity with respect to that variable.

A simple example is Newton's second law Newton's laws of motion are three physical laws that form the basis for classical mechanics. They are: of motion, which leads to the differential equation

for the motion of a particle of constant 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 distinguished from partial differential equations In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving, which involve 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 . Partial derivatives are used in vector calculus and differential geometry of several variables.

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 Sir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential people in human history. His 1687 publication of the Philosophiæ Naturalis Principia Mathematica (usually called the, Leibniz Gottfried Wilhelm Leibniz (German pronunciation: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪpnɪts]; born 1 July 1646 in Leipzig [OS: 21 June] – died in Hannover 14 November 1716) was a German mathematician and philosopher. Leibniz wrote primarily in Latin and French, the Bernoulli family, Riccati, Clairaut, d'Alembert Jean le Rond d'Alembert was a French mathematician, mechanician, physicist and philosopher. He was also co-editor with Denis Diderot of the Encyclopédie. D'Alembert's method for the wave equation is named after him and Euler Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for.

Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the, 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 Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations . This field is also known under the name numerical integration, but some people reserve this term for the computation of integrals).

The trajectory A trajectory is the path a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass. A trajectory can be described mathematically either by the geometry of the path, of a projectile A projectile is any object projected into space by the exertion of a force. Although a thrown baseball is technically a projectile too, the term more often refers to a weapon. launched from a cannon A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellants to launch a projectile. Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees, depending on their intended use on the battlefield. The follows a curve determined by an ordinary differential equation that is derived from Newton's second law.

Contents

Existence and Uniqueness of Solutions

There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. It would be an oversight to ignore the conditions which establish such existence and uniqueness. See *Picard-Lindelöf_theorem for a brief discussion of this issue.

Definitions

Ordinary differential equation

Let y be an unknown function

in x with y(n) the nth derivative In calculus 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 vehicle with respect to time is the vehicle's instantaneous velocity of y, then an equation of the form

is called an ordinary differential equation (ODE) of order n; for vector valued functions,

,

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

When a differential equation of order n has the form

it is called an implicit differential equation whereas the form

is called an explicit differential equation.

A differential equation not depending on x is called autonomous.

A differential equation is said to be linear where the differential operator L is a linear operator, y is the unknown function, and the right hand side ƒ is a given function . The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation if F can be written as a linear combination In mathematics, linear combinations is a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article of the derivatives of y

with ai(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

a function u: I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and

Given two solutions u: J ⊂ R → R and v: I ⊂ R → R, u is called an extension of v if I ⊂ J and

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 In calculus, the indefinite integral of a given function is always written with a constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function f(x) is defined on an interval and F(x) is an antiderivative of f(x), then the set of all antiderivatives of f(x) is given by. 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 In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions'. A singular solution is a solution that can't be derived from the general solution.

Examples

Main article: Examples of differential equations

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,

we define a new family of unknown functions

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

which can be written concisely in vector notation as

with

Linear ordinary differential equations

Main article: Linear differential equation where the differential operator L is a linear operator, y is the unknown function, and the right hand side ƒ is a given function . The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation

A well understood particular class of differential equations is linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1

which we can write concisely using matrix and vector notation as

with

Homogeneous equations

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

forms an n-dimensional vector space A 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 numbers, rational numbers, or even more general fields. Given a basis for this vector space , which is called a fundamental system, every solution can be written as

The n × n matrix

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.

Nonhomogeneous equations

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

can be constructed by finding the fundamental system to the corresponding homogeneous equation and one particular solution to the inhomogeneous equation. Every solution to nonhomogeneous equation can then be written as

A particular solution to the nonhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters.

Fundamental systems for homogeneous equations with constant coefficients

If a system of homogeneous linear differential equations has constant coefficients

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

with solution as a matrix exponential

which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal form

and then evaluate the Jordan blocks

of J separately as

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.

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 nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the 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.

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 Rudolf Friedrich Alfred Clebsch was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe. His collaboration with Paul Gordan in Giessen led to the introduction of Clebsch-Gordan (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.

Lie's theory

From 1870 Lie's 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 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 In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after the nineteenth century Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation 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.

Sturm-Liouville theory

Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients.

See also

Bibliography

External links

Wikibooks has a book on the topic of Calculus/Ordinary differential equations

Categories: Differential calculus | Ordinary differential equations

 

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