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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 ...
, a linear differential equation 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, ...
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 :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) 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 an unknown function of the variable . Such an equation is an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
(ODE). A ''linear differential equation'' may also be a linear
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 ...
(PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are
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 (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of
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 ...
. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any. The solutions of homogeneous linear differential equations with
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
,
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
, cosine, inverse trigonometric functions,
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...
,
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, such as computation of
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.


Basic terminology

The highest
order of derivation 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. ...
that appears in a (linear) differential equation is the ''order'' of the equation. The term , which does not depend on the unknown function and its derivatives, is sometimes called the ''constant term'' of the equation (by analogy with algebraic equations), even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be ''homogeneous'', as it is a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the ''associated homogeneous equation''. A differential equation has ''constant coefficients'' if only
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic propertie ...
s appear as coefficients in the associated homogeneous equation. A ''solution'' of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.


Linear differential operator

A ''basic differential operator'' of order is a mapping that maps any
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 ...
to its th derivative, or, in the case of several variables, to one of its
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 (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of order . It is commonly denoted :\frac in the case of univariate functions, and :\frac in the case of functions of variables. The basic differential operators include the derivative of order 0, which is the identity mapping. A linear differential operator (abbreviated, in this article, as ''linear operator'' or, simply, ''operator'') is a linear combination of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form :a_0(x)+a_1(x)\frac + \cdots +a_n(x)\frac, where are differentiable functions, and the nonnegative integer is the ''order'' of the operator (if is not the zero function). Let be a linear differential operator. The application of to a function is usually denoted or , if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
, since it maps sums to sums and the product by a scalar to the product by the same scalar. As the sum of two linear operators is a linear operator, as well as the product (on the left) of a linear operator by a differentiable function, the linear differential operators form a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or the
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 (depending on the nature of the functions that are considered). They form also a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fiel ...
over the ring of differentiable functions. The language of operators allows a compact writing for differentiable equations: if :L=a_0(x)+a_1(x)\frac + \cdots +a_n(x)\frac, is a linear differential operator, then the equation :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^=b(x) may be rewritten :Ly=b(x). There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in and the right-hand and of the equation, such as or . The ''kernel'' of a linear differential operator is its kernel as a linear mapping, that is the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of the solutions of the (homogeneous) differential equation . In the case of an ordinary differential operator of order ,
Carathéodory's existence theorem In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand sid ...
implies that, under very mild conditions, the kernel of is a vector space of dimension , and that the solutions of the equation have the form :S_0(x) + c_1S_1(x) + \cdots +c_nS_n(x), where are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval , if the functions are continuous in , and there is a positive real number such that for every in .


Homogeneous equation with constant coefficients

A homogeneous linear differential equation has ''constant coefficients'' if it has the form :a_0y + a_1y' + a_2y'' + \cdots + a_n y^ = 0 where are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients. The study of these differential equations with constant coefficients dates back to
Leonhard 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 ...
, who introduced the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, which is the unique solution of the equation such that . It follows that the th derivative of is , and this allows solving homogeneous linear differential equations rather easily. Let :a_0y + a_1y' + a_2y'' + \cdots + a_ny^ = 0 be a homogeneous linear differential equation with constant coefficients (that is are real or complex numbers). Searching solutions of this equation that have the form is equivalent to searching the constants such that :a_0e^ + a_1\alpha e^ + a_2\alpha^2 e^+\cdots + a_n\alpha^n e^ = 0. Factoring out (which is never zero), shows that must be a root of the ''characteristic polynomial'' :a_0 + a_1t + a_2t^2 + \cdots + a_nt^n of the differential equation, which is the left-hand side of the characteristic equation :a_0 + a_1t + a_2t^2 + \cdots + a_nt^n = 0. When these roots are all distinct, one has distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be linearly independent, by considering the Vandermonde determinant of the values of these solutions at . Together they form a basis of the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of solutions of the differential equation (that is, the kernel of the differential operator). In the case where the characteristic polynomial has only simple roots, the preceding provides a complete basis of the solutions vector space. In the case of
multiple root In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...
s, more linearly independent solutions are needed for having a basis. These have the form :x^ke^, where is a nonnegative integer, is a root of the characteristic polynomial of multiplicity , and . For proving that these functions are solutions, one may remark that if is a root of the characteristic polynomial of multiplicity , the characteristic polynomial may be factored as . Thus, applying the differential operator of the equation is equivalent with applying first times the operator and then the operator that has as characteristic polynomial. By the
exponential shift theorem In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (''D''-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the ''D''-operators. State ...
, :\left(\frac-\alpha\right)\left(x^ke^\right)= kx^e^, and thus one gets zero after application of As, by the fundamental theorem of algebra, the sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial, the number of above solutions equals the order of the differential equation, and these solutions form a base of the vector space of the solutions. In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of real-valued functions. Such a basis may be obtained from the preceding basis by remarking that, if is a root of the characteristic polynomial, then is also a root, of the same multiplicity. Thus a real basis is obtained by using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
, and replacing x^ke^ and x^ke^ by x^ke^ \cos(bx) and x^ke^ \sin(bx).


Second-order case

A homogeneous linear differential equation of the second order may be written :y'' + ay' + by = 0, and its characteristic polynomial is :r^2 + ar + b. If and are real, there are three cases for the solutions, depending on the discriminant . In all three cases, the general solution depends on two arbitrary constants and . * If , the characteristic polynomial has two distinct real roots , and . In this case, the general solution is ::c_1 e^ + c_2 e^. * If , the characteristic polynomial has a double root , and the general solution is ::(c_1 + c_2 x) e^. * If , the characteristic polynomial has two complex conjugate roots , and the general solution is ::c_1 e^ + c_2 e^, :which may be rewritten in real terms, using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
as :: e^ (c_1\cos(\beta x) + c_2 \sin(\beta x)). Finding the solution satisfying and , one equates the values of the above general solution at and its derivative there to and , respectively. This results in a linear system of two linear equations in the two unknowns and . Solving this system gives the solution for a so-called Cauchy problem, in which the values at for the solution of the DEQ and its derivative are specified.


Non-homogeneous equation with constant coefficients

A non-homogeneous equation of order with constant coefficients may be written :y^(x) + a_1 y^(x) + \cdots + a_ y'(x)+ a_ny(x) = f(x), where are real or complex numbers, is a given function of , and is the unknown function (for sake of simplicity, "" will be omitted in the following). There are several methods for solving such an equation. The best method depends on the nature of the function that makes the equation non-homogeneous. If is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used. If, more generally, is a linear combination of functions of the form , , and , where is a nonnegative integer, and a constant (which need not be the same in each term), then the method of undetermined coefficients may be used. Still more general, the
annihilator method In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODE's). It is similar to the method of undetermined coefficients, but instead of guess ...
applies when satisfies a homogeneous linear differential equation, typically, a holonomic function. The most general method is the variation of constants, which is presented here. The general solution of the associated homogeneous equation :y^ + a_1 y^ + \cdots + a_ y'+ a_ny = 0 is :y=u_1y_1+\cdots+ u_ny_n, where is a basis of the vector space of the solutions and are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering as constants, they can be considered as unknown functions that have to be determined for making a solution of the non-homogeneous equation. For this purpose, one adds the constraints :\begin 0 &= u'_1y_1 + u'_2y_2 + \cdots+u'_ny_n \\ 0 &= u'_1y'_1 + u'_2y'_2 + \cdots + u'_n y'_n \\ &\;\;\vdots \\ 0 &= u'_1y^_1+u'_2y^_2 + \cdots + u'_n y^_n, \end which imply (by product rule and induction) :y^ = u_1 y_1^ + \cdots + u_n y_n^ for , and :y^ = u_1 y_1^ + \cdots + u_n y_n^ +u'_1y_1^+u'_2y_2^+\cdots+u'_ny_n^. Replacing in the original equation and its derivatives by these expressions, and using the fact that are solutions of the original homogeneous equation, one gets :f=u'_1y_1^ + \cdots + u'_ny_n^. This equation and the above ones with as left-hand side form a system of linear equations in whose coefficients are known functions (, the , and their derivatives). This system can be solved by any method of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
. The computation of
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
s gives , and then . As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.


First-order equation with variable coefficients

The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of , is: :y'(x) = f(x) y(x) + g(x). If the equation is homogeneous, i.e. , one may rewrite and integrate: :\frac= f, \qquad \log y = k +F, where is an arbitrary
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
and F=\textstyle\int f\,dx is any
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of . Thus, the general solution of the homogeneous equation is :y=ce^F, where is an arbitrary constant. For the general non-homogeneous equation, one may multiply it by the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of a solution of the homogeneous equation. This gives :y'e^-yfe^= ge^. As the product rule allows rewriting the equation as :\frac\left(ye^\right)= ge^. Thus, the general solution is :y=ce^F + e^F\int ge^dx, where is a constant of integration, and is any antiderivative of (changing of antiderivative amounts to change the constant of integration).


Example

Solving the equation : y'(x) + \frac = 3x. The associated homogeneous equation y'(x) + \frac = 0 gives :\frac=-\frac, that is :y=\frac. Dividing the original equation by one of these solutions gives : xy'+y=3x^2. That is :(xy)'=3x^2, :xy=x^3 +c, and :y(x)=x^2+c/x. For the initial condition : y(1)=\alpha, one gets the particular solution :y(x)=x^2+\frac.


System of linear differential equations

A system of linear differential equations consists of several linear differential equations that involve several unknown functions. In general one restricts the study to systems such that the number of unknown functions equals the number of equations. An arbitrary linear ordinary differential equation and a system of such equations can be converted into a first order system of linear differential equations by adding variables for all but the highest order derivatives. That is, if appear in an equation, one may replace them by new unknown functions that must satisfy the equations and for . A linear system of the first order, which has unknown functions and differential equations may normally be solved for the derivatives of the unknown functions. If it is not the case this is a differential-algebraic system, and this is a different theory. Therefore, the systems that are considered here have the form :\beginy_1'(x) &= b_1(x) +a_(x)y_1+\cdots+a_(x)y_n\\ \vdots&\\ y_n'(x) &= b_n(x) +a_(x)y_1+\cdots+a_(x)y_n,\end where and the are functions of . In matrix notation, this system may be written (omitting "") :\mathbf' = A\mathbf+\mathbf. The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication. Let :\mathbf' = A\mathbf. be the homogeneous equation associated to the above matrix equation. Its solutions form a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of dimension , and are therefore the columns of a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
of functions , whose
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
is not the zero function. If , or is a matrix of constants, or, more generally, if commutes with its
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
, then one may choose equal the exponential of . In fact, in these cases, one has :\frac\exp(B) = A\exp (B). In the general case there is no closed-form solution for the homogeneous equation, and one has to use either a
numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
, or an approximation method such as
Magnus expansion In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it fu ...
. Knowing the matrix , the general solution of the non-homogeneous equation is :\mathbf(x) = U(x)\mathbf + U(x)\int U^(x)\mathbf(x)\,dx, where the column matrix \mathbf is an arbitrary
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
. If initial conditions are given as :\mathbf y(x_0)=\mathbf y_0, the solution that satisfies these initial conditions is :\mathbf(x) = U(x)U^(x_0)\mathbf + U(x)\int_^x U^(t)\mathbf(t)\,dt.


Higher order with variable coefficients

A linear ordinary equation of order one with variable coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of
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 ...
. This is not the case for order at least two. This is the main result of Picard–Vessiot theory which was initiated by
Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at ...
and
Ernest Vessiot Ernest Vessiot (; 8 March 1865 – 17 October 1952) was a French mathematician. He was born in Marseille, France, and died in La Bauche, Savoie, France. He entered the École Normale Supérieure in 1884. He was Maître de Conférences at Lill ...
, and whose recent developments are called differential Galois theory. The impossibility of solving by quadrature can be compared with the Abel–Ruffini theorem, which states that an algebraic equation of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of differential Galois theory. Similarly to the algebraic case, the theory allows deciding which equations may be solved by quadrature, and if possible solving them. However, for both theories, the necessary computations are extremely difficult, even with the most powerful computers. Nevertheless, the case of order two with rational coefficients has been completely solved by Kovacic's algorithm.


Cauchy–Euler equation

Cauchy–Euler equation In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an '' equidimensional'' equation. ...
s are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form :x^n y^(x) + a_ x^ y^(x) + \cdots + a_0 y(x) = 0, where are constant coefficients.


Holonomic functions

A holonomic function, also called a ''D-finite function'', is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients. Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s, algebraic functions,
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
,
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
,
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
, cosine, hyperbolic sine,
hyperbolic cosine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
, inverse trigonometric and
inverse hyperbolic functions In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. T ...
, and many special functions such as
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s and hypergeometric functions. Holonomic functions have several
closure properties Closure may refer to: Conceptual Psychology * Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event Computer science * Closure (computer programming), an abstraction binding a function to its scope * ...
; in particular, sums, products,
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. ...
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 ...
of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for computing the differential equation of the result of any of these operations, knowing the differential equations of the input.Zeilberger, Doron.
A holonomic systems approach to special functions identities
'. Journal of computational and applied mathematics. 32.3 (1990): 321-368
Usefulness of the concept of holonomic functions results of Zeilberger's theorem, which follows. A ''holonomic sequence'' is a sequence of numbers that may be generated by a recurrence relation with polynomial coefficients. The coefficients of 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 ...
at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
is holonomic, then the series defines a holonomic function (even if the radius of convergence is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and ''vice versa''. It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
operations can be done automatically on these functions, such as
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. ...
, indefinite and
definite integral 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 ...
, fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error,
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
s, localization of singularities, asymptotic behavior at infinity and near singularities, proof of identities, etc.Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., & Salvy, B. (2010, September).
The dynamic dictionary of mathematical functions (DDMF)
'. In International Congress on Mathematical Software (pp. 35-41). Springer, Berlin, Heidelberg.


See also

* Continuous-repayment mortgage *
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
*
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
* Linear difference equation * Variation of parameters


References

* * *


External links

* http://eqworld.ipmnet.ru/en/solutions/ode.htm
Dynamic Dictionary of Mathematical Function
Automatic and interactive study of many holonomic functions. {{DEFAULTSORT:Linear Differential Equation Differential equations