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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.
Let ˙ = be a linear first order differential equation, where () is a column vector of length and () an periodic matrix with period (that is (+) = for all real values of ). Let ϕ ( t ) {\displaystyle \phi \,(t)} be a fundamental matrix solution of this differential equation.
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
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 ...
In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the product integral solution of a first-order homogeneous linear differential equation for a linear operator.
There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kovalevskaya–Kashiwara theorem, due to Masaki Kashiwara . This theorem involves a cohomological formulation, presented in the language of D-modules.
Exact first-order ordinary differential equation; First-order differential equation; First-order differential operator; First-order linear differential equation; First-order non-singular perturbation theory; First-order partial differential equation, a partial differential equation that involves only first derivatives of the unknown function of ...
A linear differential equation that fails this condition is called inhomogeneous. A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is =