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Laser diode rate equations

The laser diode rate equations model the electrical and optical performance of a laser diode. This system of ordinary differential equations relates the number or density of photons and charge carrier

Lotka–Volterra equations

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in w

Alekseev–Gröbner formula

The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960 a

Malmquist's theorem

In mathematics, Malmquist's theorem, is the name of any of the three theorems proved by Axel Johannes Malmquist . These theorems restrict the forms of first order algebraic differential equations whic

Normal mode

A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the

Self-oscillation

Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external

Chandrasekhar's white dwarf equation

In astrophysics, Chandrasekhar's white dwarf equation is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of th

Phase plane

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate pl

Lane–Emden equation

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is n

Picone identity

In the field of ordinary differential equations, the Picone identity, named after Mauro Picone, is a classical result about homogeneous linear second order differential equations. Since its inception

Riemann's differential equation

In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere o

Caratheodory-π solution

A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. Its practicality was demo

Chandrasekhar–Page equations

Chandrasekhar–Page equations describe the wave function of the spin-½ massive particles, that resulted by seeking a separable solution to the Dirac equation in Kerr metric or Kerr–Newman metric. In 19

Painlevé transcendents

In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities a

Picard–Fuchs equation

In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.

Goldbeter–Koshland kinetics

The Goldbeter–Koshland kinetics describe a steady-state solution for a 2-state biological system. In this system, the interconversion between these two states is performed by two enzymes with opposing

Zubov's method

Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set , where is the solution to a

Separation of variables

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an

Holonomic function

In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polyn

Brusselator

The Brusselator is a theoretical model for a type of autocatalytic reaction.The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. It is a por

Frobenius method

In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form with and . in

List of nonlinear ordinary differential equations

See also List of nonlinear partial differential equations.

Carathéodory's existence theorem

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano

Oregonator

The Oregonator is a theoretical model for a type of autocatalytic reaction. The Oregonator is the simplest realistic model of the chemical dynamics of the oscillatory Belousov–Zhabotinsky reaction.It

Grothendieck–Katz p-curvature conjecture

In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogou

Generalized hypergeometric function

In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a gener

Homogeneous differential equation

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the

Sturm–Liouville theory

In mathematics and its applications, classical Sturm–Liouville theory is the theory of real second-order linear ordinary differential equations of the form: for given coefficient functions p(x), q(x),

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those function

Regge–Wheeler–Zerilli equations

In general relativity, Regge–Wheeler–Zerilli equations are a pair of equations that describes gravitational perturbations of a Schwarzschild black hole, named after Tullio Regge, John Archibald Wheele

Kneser's theorem (differential equations)

In mathematics, in the field of ordinary differential equations, the Kneser theorem, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not.

Hill differential equation

In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation where is a periodic function by minimal period . By these we mean that for all

Manakov system

Maxwell's Equations, when converted to cylindrical coordinates, and with the boundary conditions for an optical fiber while including birefringence as an effect taken into account, will yield the coup

Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.

Regular singular point

In mathematics, in the theory of ordinary differential equations in the complex plane , the points of are classified into ordinary points, at which the equation's coefficients are analytic functions,

Annihilator method

In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODE's). It is similar to the method of und

Isomonodromic deformation

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 nonlin

Michaelis–Menten kinetics

In biochemistry, Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes

Characteristic multiplier

In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as char

Peano existence theorem

In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy,

Lagrange's identity (boundary value problem)

In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration

Riemann–Hilbert problem

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existe

Schwarzian derivative

In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in

Arditi–Ginzburg equations

The Arditi–Ginzburg equations describes ratio dependent predator–prey dynamics. Where N is the population of a prey species and P that of a predator, the population dynamics are described by the follo

Atkinson–Mingarelli theorem

In applied mathematics, the Atkinson–Mingarelli theorem, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators. In the s

Growth elasticity of poverty

Growth elasticity of poverty (GEP) is the percentage reduction in poverty rates associated with a percentage change in mean (per capita) income. Mathematically; where PR is a poverty measure and y is

Laplace transform applied to differential equations

In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear d

Baer function

Baer functions and , named after Karl Baer, are solutions of the Baer differential equation which arises when separation of variables is applied to the Laplace equation in paraboloidal coordinates. Th

Growth curve (statistics)

The growth curve model in statistics is a specific multivariate linear model, also known as GMANOVA (Generalized Multivariate Analysis-Of-Variance). It generalizes MANOVA by allowing post-matrices, as

Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or

Ordered exponential

The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative alg

Parasitic oscillation

Parasitic oscillation is an undesirable electronic oscillation (cyclic variation in output voltage or current) in an electronic or digital device. It is often caused by feedback in an amplifying devic

Exponential integrator

Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This large class of methods from numerical analysis i

Soil moisture velocity equation

The soil moisture velocity equation describes the speed that water moves vertically through unsaturated soil under the combined actions of gravity and capillarity, a process known as infiltration. The

Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to

Dynamic simulation

Dynamic simulation (or dynamic system simulation) is the use of a computer program to model the time-varying behavior of a dynamical system. The systems are typically described by ordinary differentia

Mingarelli identity

In the field of ordinary differential equations, the Mingarelli identity is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations

Magnus expansion

In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equati

PECE

PECE is a technique of handling an implicit ordinary differential equation approximation formula by a prediction (P) step and a single correction (C) step. (The E's represent evaluations of the deriva

Phase line (mathematics)

In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, . The phase line is the 1-dimensional form of the g

Perturbation theory

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A cri

Finite water-content vadose zone flow method

The finite water-content vadose zone flux method represents a one-dimensional alternative to the numerical solution of Richards' equation for simulating the movement of water in unsaturated soils. The

Abel equation of the first kind

In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the

Bernoulli differential equation

In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form where is a real number. Some authors allow any real , whereas others require that not

Characteristic equation (calculus)

In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equati

Monodromy matrix

In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It

Variation of parameters

In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear d

Wronskian

In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by Józef Hoene-Wroński and named by Thomas Muir . It is used in the study of differential equations, where it can sometimes sho

Inexact differential equation

An inexact differential equation is a differential equation of the form (see also: inexact differential) The solution to such equations came with the invention of the integrating factor by Leonhard Eu

Method of undetermined coefficients

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closel

Mathieu function

In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation where and are parameters. They were first introduced by Émile Léonard Ma

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerica

Emden–Chandrasekhar equation

In astrophysics, the Emden–Chandrasekhar equation is a dimensionless form of the Poisson equation for the density distribution of a spherically symmetric isothermal gas sphere subjected to its own gra

Sturm–Picone comparison theorem

In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides

Heun function

In mathematics, the local Heun function Hℓ(a,q;α,β,γ,δ;z) (Karl L. W. Heun ) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun func

Fundamental matrix (linear differential equation)

In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations is a matrix-valued function whose columns are linearly independent solutions of the system.Then

Spheroidal wave equation

In mathematics, the spheroidal wave equation is given by It is a generalization of the Mathieu differential equation.If is a solution to this equation and we define , then is a prolate spheroidal wave

Node (autonomous system)

The behaviour of a linear autonomous system around a critical point is a node if the following conditions are satisfied: Each path converges to the or away from the critical point (dependent of the un

Picard–Lindelöf theorem

In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard'

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

Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations,

Halanay inequality

Halanay inequality is a comparison theorem for differential equations with delay. This inequality and its generalizations have been applied to analyze the stability of delayed differential equations,

Sturm separation theorem

In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous secon

Abel's identity

In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear

Green's matrix

In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named

Chrystal's equation

In mathematics, Chrystal's equation is a first order nonlinear ordinary differential equation, named after the mathematician George Chrystal, who discussed the singular solution of this equation in 18

Frobenius solution to the hypergeometric equation

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that us

Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: where k is a positive

Spectral theory of ordinary differential equations

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a lin

Parametric oscillator

A parametric oscillator is a driven harmonic oscillator in which the oscillations are driven by varying some parameter of the system at some frequency, typically different from the natural frequency o

Van der Pol oscillator

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: where x is the position coordi

Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the ene

Euler's differential equation

In mathematics, Euler's differential equation is a first order nonlinear ordinary differential equation, named after Leonhard Euler given by This is a separable equation and the solution is given by t

Cauchy–Euler equation

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 referr

Exact differential equation

In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.

Exponential response formula

In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary

Oscillation theory

In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation is called oscillating if it has an infinite number of roots; otherwise it is

Adams–Williamson equation

The Adams–Williamson equation, named after Leason H. Adams and E. D. Williamson, is an equation used to determine density as a function of radius, more commonly used to determine the relation between

Meissner equation

The Meissner equation is a linear ordinary differential equation that is a special case of Hill's equation with the periodic function given as a square wave. There are many ways to write the Meissner

D'Alembert's equation

In mathematics, d'Alembert's equation is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as where . After diffe

Burchnall–Chaundy theory

In mathematics, the Burchnall–Chaundy theory of commuting linear ordinary differential operators was introduced by Burchnall and Chaundy . One of the main results says that two commuting differential

Euler–Lagrange equation

In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given acti

Isocline

Given a family of curves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains a given slope. The word comes from the Gre

Ince equation

In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation When p is a non-negative integer, it has polynomial solutions called Ince polynomials.

Monod equation

The Monod equation is a mathematical model for the growth of microorganisms. It is named for Jacques Monod (1910 – 1976, a French biochemist, Nobel Prize in Physiology or Medicine in 1965), who propos

Power series solution of differential equations

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then su

Movable singularity

In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the

Rayleigh–Plesset equation

In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is an ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incom

Bateman equation

In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model

COPASI

COPASI (COmplex PAthway SImulator) is an open-source software application for creating and solving mathematical models of biological processes such as metabolic networks, cell-signaling pathways, regu

Matrix differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. A matrix

Chebyshev equation

Chebyshev's equation is the second order linear differential equation where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be

Chazy equation

In mathematics, the Chazy equation is the differential equation It was introduced by Jean Chazy as an example of a third-order differential equation with a movable singularity that is a natural bounda

Exponential growth

Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the q

Binomial differential equation

In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions. For example

Hilbert's twenty-first problem

The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specif

Bony–Brezis theorem

In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant

Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and

Liñán's equation

In the study of diffusion flame, Liñán's equation is a second-order nonlinear ordinary differential equation which describes the inner structure of the diffusion flame, first derived by Amable Liñán i

Riccati equation

In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form whe

Thomas–Fermi equation

In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi, which can be derived by apply

Reduction of order

Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution is known and a second linearly independent solution i

Method of dominant balance

In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterati

Duffing equation

The Duffing equation (or Duffing oscillator), named after (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

Clairaut's equation (mathematical analysis)

In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form where f is continuously differentiable. It is a particular case of the Lagrange differen

Grönwall's inequality

In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or by the solutio

Autonomous system (mathematics)

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable

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