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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, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems.
In the theory of linear differential equations with one independent variable z, I wish to indicate an important problem one which very likely Riemann himself may have had in mind. This problem is as follows: To show that there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group.
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., for a form to be exact, it ...
The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms.
It should only contain pages that are Linear (group) songs or lists of Linear (group) songs, as well as subcategories containing those things (themselves set categories). Topics about Linear (group) songs in general should be placed in relevant topic categories .
In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system.
Nicholas Katz has applied Tannakian category techniques to show that this conjecture is essentially the same as saying that the differential Galois group G (or strictly speaking the Lie algebra g of the algebraic group G, which in this case is the Zariski closure of the monodromy group) can be determined by mod p information, for a certain wide class of differential equations.