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  2. Linear differential equation - Wikipedia

    en.wikipedia.org/wiki/Linear_differential_equation

    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.

  3. Floquet theory - Wikipedia

    en.wikipedia.org/wiki/Floquet_theory

    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.

  4. System of differential equations - Wikipedia

    en.wikipedia.org/wiki/System_of_differential...

    For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:

  5. Numerical methods for ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Numerical_methods_for...

    First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.

  6. Magnus expansion - Wikipedia

    en.wikipedia.org/wiki/Magnus_expansion

    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.

  7. Ordinary differential equation - Wikipedia

    en.wikipedia.org/wiki/Ordinary_differential_equation

    Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function). When physical phenomena are modeled with non-linear equations, they ...

  8. Euler method - Wikipedia

    en.wikipedia.org/wiki/Euler_method

    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. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta ...

  9. Liouville's formula - Wikipedia

    en.wikipedia.org/wiki/Liouville's_formula

    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.