Differential equation
Given that y is a function of x and that
When a differential equation of order n has the form
A differential equation not depending on x is called autonomous.
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
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.
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.
The first method of integrating linear ordinary differential
equations with constant coefficients is due to Euler, who made the
solution of the form
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.
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 analyic 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 equations) 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. Since about 1870, Darboux, Kovalevsky, Méray,
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).
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. 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.
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 (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.
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.
Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently 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
f = 0 under rational one-to-one transformations.
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).
Definition
denote the derivatives
an ordinary differential equation (ODE) is an equation involving
The order of a differential equation is the order of the highest derivative that appears.
it is called an implicit differential equation whereas the form
is called an explicit differential equation.General application
has the general solution
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).History
Linear ODEs with constant coefficients
depend on that of the algebraic equation of the nth degree,
in which zk takes the place of
This equation F(z) = 0, is the "characteristic"
equation considered later by Monge and Cauchy.Linear PDEs
First-order PDEs
Singular solutions
Reduction to quadratures
The Fuchsian theory
Lie's theory
See also
| Topics in mathematics related to change | Edit |
| Arithmetic | Calculus | Vector calculus | Analysis | Differential equations | Dynamical systems and chaos theory | List of functions |