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Home > Differential equation

1 Definition

Given that y is a function of x and that

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.

When a differential equation of order n has the form

it is called an implicit differential equation whereas the form

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.

2 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

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).

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 dynamicsFluid dynamics is the study of fluids ( liquids and gases) in motion, and the effect of the fluid motion on fluid boundaries, such as solid containers or other fluids. Fluid dynamics is a branch of fluid mechanics, and has a number of subdisciplines, incl or celestial mechanicsCelestial Mechanics Astrodynamics Celestial mechanics is a term for the application of physics, historically Newtonian mechanics, to astronomical objects such as stars and planets. After Einstein explained the anomalous precession of Mercury's perihelion,. Therefore, the study of differential equations is a wide field in both pure and applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations.

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.

3 Types of differential equations with some history

The influence of geometry, physics, and astronomy, starting with NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis and LeibnizGottfried Wilhelm von Leibniz ( July 1, 1646 in Leipzig November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. Leibniz is credited with the term " function" ( 1694), which he use, 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 .

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Topics: Differential Equation Equations Theory Partial Solutions...

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