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Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation , Parabolic partial differential equation and Hyperbolic partial differential equation .
The exact solution of the differential equation is () =, so () =. Although the approximation of the Euler method was not very precise in this specific case, particularly due to a large value step size h {\displaystyle h} , its behaviour is qualitatively correct as the figure shows.
In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues for differential equations of algebraically closed ...
Binomial differential equation (′) = (,) Class of differential equation which may sometimes be solved exactly [3] Briot-Bouquet Equation: 1 ′ = (,) Class of differential equation which may sometimes be solved exactly [4]
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
The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term b(x), which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function.