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Differential equation

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In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. In practice the "unknown function" is usually presumed to exist, although rigorously establishing this may require techniques from topology. The order of a differential equation is given by the maximum number of times the supposed unknown function in it has been differentiated. See differential calculus and integral calculus for basic calculus background.

Contents

1 Definition

2 General application

3 History

3.1 Linear ODEs with constant coefficients
3.2 Linear PDEs
3.3 First-order PDEs
3.4 Singular solutions
3.5 Reduction to quadratures
3.6 The Fuchsian theory
3.7 Lie's theory

4 See also

Definition

Given that y is a function of x and that

$y', y'',\ \dots,\ y^{(n)}$

denote the derivatives

$\frac{dy}{dx},\ \frac{d^{2}y}{dx^2},\ \dots,\ \frac{d^{n}y}{dx^{n}},$

an ordinary differential equation (ODE) is an equation involving

$x,\ y,\ y',\ y'',\ \dots$ .

The order of a differential equation is the order $n$ of the highest derivative that appears.

When a differential equation of order n has the form

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

it is called an implicit differential equation whereas the form

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

is called an explicit differential equation.

A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

General application

An important special case is when the equations do not involve $x$ . These differential equations may be represented as vector fields. This type of differential equations has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also symplectic topology for abstract discussion.)

The problem of solving a differential equation is to find the function $y$ whose derivatives satisfy the equation. For example, the differential equation

$y'' + y = 0 \, \!$

has the general solution

$y = A \cos{x} + B \sin{x} \, \!$ ,

where A, B are constants determined from boundary conditions. In the case where the equations are linear, this can be done by breaking the original equation down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see numerical ordinary differential equations).

Ordinary differential equations are to be distinguished from partial differential equations where $y$ is a function of several variables, and the differential equation involves partial derivatives.

Differential equations are used to construct mathematical models of physical phenomena such as fluid dynamics or celestial mechanics. Therefore, the study of differential equations is a wide field in both pure and applied mathematics.

Differential equations have intrinsically interesting properties such as whether or not solutions exist, and should solutions exist, whether those solutions are unique. Applied mathematicians, physicists and engineers are usually more interested in how to compute solutions to differential equations. These solutions are then used to design bridges, automobiles, aircraft, sewers, etc.

History

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients .

Linear ODEs with constant coefficients

The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who made the solution of the form

$\frac {d^{n}y} {dx^{n}} + A_{1}\frac {d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = 0$

depend on that of the algebraic equation of the nth degree,

$F(z) = z^{n} + A_{1}z^{n-1} + \cdots + A_n = 0$

in which z^k takes the place of

$\frac {d^{k}y} {dx^{k}}\quad\quad(k = 1, 2, \cdots, n).$

This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy.

If z is a (possibly complex) zero of F(z) of multiplicity m and $k\in\{0,1,\dots,m-1\}$ then $y = x k e z x$ is a solution of the ODE.

If the A_i are real then real-valued solutions are preferable. Since the complex zs will come in conjugate pairs, so will their ys; replace each pair with their linear combos $\Re y$ and $\Im y$ .

Example: ODE, $y'''' - 2 y''' + 2 y'' - 2 y' + y = 0$ . Characteristic eq'n, $z 4 - 2 z 3 + 2 z 2 - 2 z + 1 = 0$ . Zeroes, i, −i, 1 (multiplicity 2). Solution basis, $e i x$ , $e - i x$ , $e x$ , $x e x$ . Real-valued solution basis, $cos x$ , $sin x$ , $e x$ , $x e x$ .

Linear PDEs

The theory of linear partial differential equations may be said to begin with Lagrange (1779 to 1785). Monge (1809) treated ordinary and partial differential equations of the first and second order, uniting the theory to geometry, and introducing the notion of the "characteristic", the curve represented by $F (z) = 0$ , which was investigated by Darboux , Levy, and Lie.

First-order PDEs

Pfaff (1814, 1815) gave the first general method of integrating partial differential equations of the first order, of which Gauss (1815) gave an analysis. Cauchy (1819) gave a simpler method, attacking the subject from the analytical standpoint, but using the Monge characteristic . Cauchy also first stated the theorem (now called the Cauchy-Kowaleskaya theorem ) that every analytic differential equation defines an analytic function, expressible by means of a convergent series.

Jacobi (1827) also gave an analysis of Pfaff's method, besides developing an original one (1836) which Clebsch published (1862). Clebsch's own method appeared in 1866, and others are due to Boole (1859), Korkine (1869), and A. Mayer (1872). Pfaff's problem (on total differential equation s) was investigated by Natani (1859), Clebsch (1861, 1862), DuBois-Reymond (1869), Cayley, Baltzer, Frobenius, Morera, Darboux, and Lie.

The next great improvement in the theory of partial differential equations of the first order was made by Lie (1872), who placed the whole subject on a solid foundation. After about 1870, Darboux, Kovalevsky, Méray, Mansion, Graindorge, and Imschenetsky became prominent in this line. The theory of partial differential equations of the second and higher orders, beginning with Laplace and Monge, was notably advanced by Ampère (1840).

The integration of partial differential equations with three or more variables was the object of elaborate investigations by Lagrange, and his name became connected with certain subsidiary equations. It was he and Charpit who originated one of the methods for integrating the general equation with two variables; a method which now bears Charpit's name.

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 $n$ 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.

The 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 integral s. 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 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 Lie groups (as they are now called) 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).

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