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An ordinary differential equation ( ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y ), which, therefore, depends on x. Thus x is often called the independent variable of the equation.
In mathematics, an ordinary differential equation ( ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown (s) consists of one (or more) function (s) and involves the derivatives of those functions. [1] The term "ordinary" is used in contrast with partial differential equations (PDEs ...
The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of y′ (x), is: If the equation is homogeneous, i.e. g(x) = 0, one may rewrite and integrate: where k is an arbitrary constant of integration and is any antiderivative of f.
Cauchy–Euler equation. In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential ...
Matrix differential equation. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to ...
An exact equation may also be presented in the following form: where the same constraints on I and J apply for the differential equation to be exact. The nomenclature of "exact differential equation" refers to the exact differential of a function. For a function , the exact or total derivative with respect to is given by.
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written. where f and g are homogeneous functions of the same degree of x and y. [1] In this case, the change of variable y = ux leads to an equation of the form. which is easy to solve by integration ...
Differential equations. In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form. where is a real number. Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [3] [4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named.
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