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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 equation can be solved ...
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. [1] A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy.
Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow.
Lagrange's identity (boundary value problem) Lane–Emden equation; Laser diode rate equations; Legendre's differential equation; Libroadrunner; Liénard equation; Liñán's equation; Liouville's formula; Lommel function; Lotka–Volterra equations; Lumped parameter model for the cardiovascular system
In mathematics, a Cauchy (French:) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the ...