Ordinary differential equation

Overview

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an

**ordinary differential equation**(or

**ODE**) is a relation that contains functions of only one independent variable

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...

, and one or more of their derivative

Derivative

In calculus, a branch of mathematics, 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...

s with respect to that variable.

A simple example is Newton's second law of motion, which leads to the differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

for the motion of a particle of constant mass m.

Discussions

Encyclopedia

In mathematics

, an

, and one or more of their derivative

s with respect to that variable.

A simple example is Newton's second law 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 x(t) of the particle 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 equation

s, which involve partial derivative

s of functions of several variables.

Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many mathematicians have studied differential equations and contributed to the field, including Newton

, Leibniz

, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.

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 x with the nth derivative

of y, and let F be a given function

then an equation of the form

is called an

it is called a

More generally, an

where F : ℝ

is called an

A differential equation not depending on x is called

A differential equation is said to be

of the derivatives of y together with a constant term, all possibly depending on x:

with a

a function is called the

for F, if u is n-times differentiable on I, and

Given two solutions and , u is called an

A solution which has no extension is called a

A

or boundary conditions

'. A singular solution

is a solution that can't be derived from the general solution.

and the Peano existence theorem

.

Given an explicit ordinary differential equation of order n (and dimension 1),

define a new family of unknown functions

for i from 1 to n.

The original differential equation can be rewritten as the system of differential equations with order 1 and dimension n given by

which can be written concisely in vector notation as

with

and

which we can write concisely using matrix and vector notation as

with

forms an n-dimensional vector space

. Given a basis for this vector space , which is called a

The n × n matrix

is called

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

.

Concerning second order linear ordinary differential equations, it is well known that

So, if is a solution of: , then such that:

So, if is a solution of: ; then a particular solution of , is given by: .

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

and then evaluate the Jordan blocks

of J separately as

let U(

If A(x

may have to be used.

s 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.

s. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the 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 number

s 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.

(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.

'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 group

s, be referred to a common source; and that

ordinary differential equations which admit the same infinitesimal transformation

s present comparable difficulties of integration. He

also emphasized the subject of transformations of contact.

A general approach to solve DE's uses the symmetry property of differential equations, the continuous infinitesimal transformation

s of solutions to solutions (Lie theory

). Continuous group theory

, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pair

s, recursion operators, Bäcklund transform

and finally finding exact analytic solutions to the DE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

operators defined in terms of second-order homogeneous linear equations, and is useful

in the analysis of certain partial differential equations.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an

**ordinary differential equation**(or**ODE**) is a relation that contains functions of only one independent variableIndependent 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...

, and one or more of their derivative

Derivative

In calculus, a branch of mathematics, 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...

s with respect to that variable.

A simple example is Newton's second law of motion, which leads to the differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

for the motion of a particle of constant mass m. In general, the force F depends upon the position x(t) of the particle 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 equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s, which involve partial derivative

Partial derivative

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...

s of functions of several variables.

Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many mathematicians have studied differential equations and contributed to the field, including Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, Leibniz

Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.

Much study has been devoted to the solution of ordinary differential equations.

In the case where the equation is linear

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

, 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

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...

).

### Ordinary differential equation

Let y be an unknown functionin x with the nth derivative

Derivative

In calculus, a branch of mathematics, 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...

of y, and let F be a given function

then an equation of the form

is called an

**ordinary differential equation (ODE)**of**order**n. If y is an unknown vector valued function,it is called a

**system of ordinary differential equations**of**dimension**m (in this case, F : ℝ^{mn+1}→ ℝ^{m}).More generally, an

**implicit**ordinary differential equation of order n has the formwhere F : ℝ

^{n+2}→ ℝ depends on y^{(n)}. To distinguish the above case from this one, an equation of the formis called an

**explicit**differential equation.A differential equation not depending on x is called

**autonomous**

.Autonomous system (mathematics)

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

A differential equation is said to be

**linear**

if F can be written as a linear combinationLinear differential equation

Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

of the derivatives of y together with a constant term, all possibly depending on 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 equationa function is called the

**solution**or integral curveIntegral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...

for F, if u is n-times differentiable on I, and

Given two solutions and , u is called an

**extension**of v if andA 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 conditionsInitial value problem

In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

or boundary conditions

Boundary value problem

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 singular solution

Singular solution

A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...

is a solution that can't be derived from the general solution.

## 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. The two main theorems are the Picard–Lindelöf theoremPicard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.The theorem is named after Charles...

and the Peano existence theorem

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 the existence of solutions to certain initial value...

.

## 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),

define a new family of unknown functions

for i from 1 to n.

The original differential equation can be rewritten as the system of differential equations with order 1 and dimension n given by

which can be written concisely in vector notation as

with

and

## Linear ordinary differential equations

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 equations of order 1which 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 nforms an n-dimensional vector space

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...

. Given a basis for this vector space , which is called a

**fundamental system**, every solution can be written asThe 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 ncan 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

Method of undetermined coefficients

In mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...

or the method of variation of parameters

Method of variation of parameters

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations...

.

Concerning second order linear ordinary differential equations, it is well known that

So, if is a solution of: , then such that:

So, if is a solution of: ; then a particular solution of , is given by: .

### Fundamental systems for homogeneous equations with constant coefficients

If a system of homogeneous linear differential equations has constant coefficientsthen we can explicitly construct a fundamental system. The fundamental system can be written as a matrix differential equation

Matrix differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders...

with solution as a matrix exponential

Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....

which is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform

**A**into Jordan normal formJordan normal form

In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...

and then evaluate the Jordan blocks

of J separately as

### General Case

To solve**y**'(*x*) = A(*x*)**y**(*x*)+**b**(*x*) with**y**(x_{0}) =**y**(here_{0}**y**(*x*) is a vector or matrix, and A(*x*) is a matrix),

let U(

*x*) be the solution of**y**'(*x*) = A(*x*)**y**(x) with U(x_{0}) = I (the identity matrix). After substituting**y**(*x*) = U(*x*)**z**(*x*), the equation**y**'(*x*) = A(*x*)**y**(*x*)+**b**(*x*) simplifies to U(*x*)z'(*x*) = B(*x*). Thus,If A(x

_{1}) commutes with A(x_{2}) for all x_{1}and x_{2}, then (and thus ), but in the general case there is no closed form solution, and an approximation method such as Magnus expansionMagnus expansion

In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus , provides an exponential representation of the solution of a first order linear homogeneous differential equation for a linear operator. In particular it furnishes the fundamental matrix of a system of linear ordinary...

may have to be used.

### Singular solutions

The theory of singular solutionSingular solution

A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...

s 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

Jean Gaston Darboux

Jean-Gaston Darboux was a French mathematician.-Life:Darboux made several important contributions to geometry and mathematical analysis . He was a biographer of Henri Poincaré and he edited the Selected Works of Joseph Fourier.Darboux received his Ph.D...

(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

Felice Casorati (mathematician)

Felice Casorati was an Italian mathematician best known for the Casorati-Weierstrass theorem in complex analysis...

and Cayley

Arthur Cayley

Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

. 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 quadratureQuadrature (mathematics)

Quadrature — historical mathematical term which means calculating of the area. Quadrature problems have served as one of the main sources of mathematical analysis.- History :...

s. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the th degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

(1799) showed, however, that the differential equation meets its limitations very soon unless complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s 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 FuchsLazarus Fuchs

Lazarus Immanuel Fuchs was a German mathematician who contributed important research in the field of linear differential equations...

(Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius

Ferdinand Georg Frobenius

Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...

. Collet was a prominent contributor beginning in 1869, although his method for integrating a

non-linear system was communicated to Bertrand in 1868. Clebsch

Alfred 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...

(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 Sophus LieSophus Lie

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

'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 group

Lie group

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...

s, be referred to a common source; and that

ordinary differential equations which admit the same infinitesimal transformation

Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...

s present comparable difficulties of integration. He

also emphasized the subject of transformations of contact.

A general approach to solve DE's uses the symmetry property of differential equations, the continuous infinitesimal transformation

Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...

s of solutions to solutions (Lie theory

Lie theory

Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....

). Continuous group theory

Group theory

In mathematics and 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 as groups endowed with additional operations and...

, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pair

Lax pair

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They were introduced by Peter Lax to discuss solitons in continuous media...

s, recursion operators, Bäcklund transform

Bäcklund transform

In mathematics, Bäcklund transforms or Bäcklund transformations relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems...

and finally finding exact analytic solutions to the DE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

### Sturm–Liouville theory

Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linearoperators defined in terms of second-order homogeneous linear equations, and is useful

in the analysis of certain partial differential equations.

## See also

- Numerical ordinary differential equationsNumerical ordinary differential equationsNumerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...
- Difference equation
- Matrix differential equationMatrix differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders...
- Laplace transform applied to differential equations
- Boundary value problemBoundary value problemIn 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...
- List of dynamical systems and differential equations topics
- Separation of variablesSeparation of variablesIn mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
- Method of undetermined coefficientsMethod of undetermined coefficientsIn mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...

## External links

(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 UniversityLamar UniversityLamar University, often referred to as Lamar or LU, is a comprehensive coeducational public research university located in Beaumont, Texas, United States. Lamar confers bachelors, masters and doctoral degrees and is classified as a Doctoral Research University by the Carnegie Commission on Higher...

. - Differential Equations, S.O.S. Mathematics.
- A primer on analytical solution of differential equations from the Holistic Numerical Methods Institute, University of South Florida.
- Ordinary Differential Equations and Dynamical Systems lecture notes by Gerald TeschlGerald TeschlGerald Teschl is an Austrian mathematical physicist and Professor of Mathematics.He is working in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable partial differential equations .-Career:After studying physics at the Graz...

. - Notes on Diffy Qs: Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC.