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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.
Definition. Given a simply connected and open subset D of and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form. is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, [1] [2] so that.
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form + ′ + ″ + () = where a 0 (x), ..., a n (x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y (n) are the successive derivatives of an unknown function y of ...
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 ...
Calculus. In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. [1] It is one of the two traditional divisions of calculus, the other being integral calculus —the study of the area beneath a curve. [2]
In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold. Indeed, let f : U → R r {\displaystyle f:U\to \mathbb {R} ^{r}} be such a smooth map from an open subset of R n {\displaystyle \mathbb {R} ^{n}} (since the result is local, there is no loss of ...
Boussinesq approximation (buoyancy) Boussinesq approximation (water waves) Buckley–Leverett equation. Camassa–Holm equation. Chaplygin's equation. Continuity equation for conservation laws. Convection–diffusion equation. Double diffusive convection. Davey–Stewartson equation.
A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields. For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., a form to be exact, it needs to ...