Differential equation
Given that y is a function of x and that
- .
General application
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.
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
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, 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 was 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).
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.
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 Fuchsian theory
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 transformation]]s present comparable difficulties of integration. He also emphasized the subject of transformations of contact (Berührungstransformationen).
See also
- Examples of differential equations
- Differential equations of mathematical physics
- Differential equations from outside physics
- Difference equation
- Laplace transform applied to differential equations
- Boundary value problem
- List of dynamical system and differential equation topics
Topics in mathematics related to change |
Arithmetic | Calculus | Vector calculus | Analysis | Differential equations | Dynamical systems and chaos theory | Fractional calculus | List of functions |