Ordinary differential equation
In mathematics, and more specifically calculus, a differential equation is an equation that describes the relationship between an unknown function and its derivatives. The order of a differential equation describes the most times any function in it has been differentiated. (See differential calculus and integral calculus.)
Given that  is a function of  and that
denote the derivatives
 ,
an ordinary differential equation is an equation involving
 .
The order of a differential equation is the order  of the highest derivative that appears.
An important special case is when the equations do not involve . 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 whose derivatives satisfy the equation. For example, the differential equation  has the general solution  , where  ,  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 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 his type,  , depend on that of the
algebraic equation of the
th degree,  , in
which  takes the place of  . This equation  , is the "characteristic"
equation considered later by Monge and Cauchy.
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 , which has
recently been 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, a method of which Gauss
(1815) at once recognized the value and of which he gave an
analysis. Soon after, Cauchy (1819) gave a simpler method, attacking
the subject from the analytical standpoint, but using the Monge
characteristic. To him is also due the theorem, corresponding to the
fundamental theorem of algebra, that every differential equation
defines a function expressible by means of a convergent series, a
proposition more simply proved by Briot and Bouquet, and also by Picard (1891). 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 has been a prominent subject of
investigation, and with it are connected the names of 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 is
due to Lie (1872), by whom the whole subject was placed on a
solid foundation. Since about 1870, Darboux, Kovalevsky, M\'eray,
Mansion, Graindorge, and Imschenetsky have been 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). Imschenetsky has summarized the contributions to
1873, but the theory remained in an imperfect state.
The integration of partial differential equations with three or more
variables was the object of elaborate investigations by Lagrange,
and his name is still connected with certain subsidiary
equations. To him and to Charpit, who did much to develop the
theory, is due 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 especial attention. A valuable but little-known work on the
subject is that of Houtain (1854). Darboux (from 1873) has been a
leader in the theory, and in the geometric interpretation of these
solutions he has opened a field which has been 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 at present accepted.
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 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, and the modern
theory may be said to begin with him. 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
A departure took its inspiration from two memoirs by Fuchs (Crelle, 1866, 1868), a work
elaborated by Thomé and Frobenius. Collet has been a prominent
contributor since 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
 under rational one-to-one transformations.
Lie's theory
From 1870 Lie's labors put the entire theory of differential equations
on a more satisfactory foundation. He showed that the integration
theories of the older mathematicians, which had been looked upon as
isolated, can by the introduction of the concept of continuous
groups of transformations be referred to a common source, and that
ordinary differential equations which admit the same infinitesimal
transformations present like difficulties of integration. He
also emphasized the subject of transformations of contact
(Berührungstransformationen) which underlies so much of the recent
theory. The modern school has also turned its attention to the
theory of differential invariants, one of fundamental importance and
one which Lie has made prominent. With this theory are associated
the names of Cayley, Cockle, Sylvester, Forsyth, Laguerre, and
Halphen. Recent writers have shown the same tendency noticeable in
the work of Monge and Cauchy, the tendency to separate into two
schools, the one inclining to use the geometric diagram, and
represented by Schwarz, Klein, and Goursat, the other adhering to
pure analysis, of which Weierstrass, Fuchs, and Frobenius are
types. The work of Fuchs and the theory of elementary divisors have
formed the basis of a late work by Sauvage (1895). Poincar\'e's
recent contributions are also very notable. His theory of Fuchsian equations (also investigated by Klein) is connected with the general
theory. He has also brought the whole subject into close relations
with the theory of functions. Appell has recently contributed to the
theory of linear differential equations transformable into
themselves by change of the function and the variable. Helge von
Koch has written on infinite determinants and linear differential
equations. Picard has undertaken the generalization of the work of
Fuchs and Poincar\'e in the case of differential equations of the
second order. Fabry (1885) has generalized the normal integrals of
Thomé, integrals which Poincar\'e has called "intégrales
anormales," and which Picard has recently studied. Riquier
treated the question of the existence of integrals in any
differential system and gave a brief summary of the history to
1895. The later contributors include Brioschi,
Königsberger, Peano, Graf, Hamburger, Graindorge, Schläfli,
Glaisher, Lommel, Gilbert, Fabry, Craig, and Autonne.
See also
Referenced By
Laplace transform applied to differential equations | List of dynamical system and differential equation topics | List of dynamical system topics | List of mathematical topics (M-O)
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