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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 ...
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients.
Identifying first order exact differential equations. Let the functions , , , and , where the subscripts denote the partial derivative with respect to the relative variable, be continuous in the region . Then the differential equation. is exact if and only if. That is, there exists a function , called a potential function, such that.
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 ...
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.
In mathematics, a differential-algebraic system of equations ( DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential algebra of differential ...
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