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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.
e. 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 ...
Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear 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.
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point. 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 ...
The same illustration for The midpoint method converges faster than the Euler method, as . Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to ...
Definition. An autonomous system is a system of ordinary differential equations of the form where x takes values in n -dimensional Euclidean space; t is often interpreted as time. It is distinguished from systems of differential equations of the form in which the law governing the evolution of the system does not depend solely on the system's ...
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.