Ordinary differential equations

differential equation containing one or more functions of one independent variable and its derivatives

physical system that responds to a restoring force inversely proportional to displacement

growth of quantities at rate proportional to the current amount

thumb|upright|Underdamped spring–mass system with

second-order partial differential equation whose solutions are the functions for which a given functional is stationary

Model of enzyme kinetics

System of ordinary differential equations first studied by Edward Lorenz

type of ordinary differential equation

first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey

mathematical methods used to find an approximate solution to a problem which cannot be solved exactly

differential equation together with a set of additional constraints (boundary conditions)

pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation

In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order . It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.

type of differential equation

method

function that is commonly used to solve ordinary differential equations

linear homogeneous ordinary differential equation

theorem on existence and uniqueness of solutions to first-order equations with given initial conditions

ordinary differential equation

special function defined by a hypergeometric series

thumb|right|300px|Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y' = xy. The solution curves are y = C e^{x^2/2}. 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 Greek words ἴσος (isos), meaning "same", and the κλίνειν (klenein), meaning "make to slope". Generally, an isocline will itself have the shape of a curve or the union of a small number of curves.

mathematical relation with derivatives

special function in the physical sciences

Method for solving ordinary differential equations

procedure for solving differential equations

type of differential equation

non-conservative oscillator with non-linear damping

dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid

mathematical equations

curve that is a parametric solution to an initial-value problem given by a vector field

methods used to find numerical solutions of ordinary differential equations

theorem that gives bounds on integrals of functions

second order linear differential equation featuring a periodic function

thumb|300px|Schematic representation of a self-oscillation as a positive feedback loop. The oscillator V produces a feedback signal B. The controller at R uses this signal to modulate the external power S that acts on the oscillator. If the power is modulated in phase with the oscillator's velocity, a negative damping is established and the oscillation grows until limited by nonlinearities.

theorem

theory of 2nd‐order linear ODEs that are eigenvalue equations of the operator 𝑤(𝑥)⁻¹((d∕d𝑥)𝑝(𝑥)d∕d𝑥+𝑞(𝑥))

approach for finding solutions of nonhomogeneous ordinary differential equations

solutions to Mathieu's differential equations, which are second order linear differential equation

harmonic oscillator whose parameters oscillate in time

algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation

non-linear second order differential equation and its attractor

visual representation used in non-linear control system analysis

Special functions in mathematics

family of power series in mathematics

thumb|right|350px|Top: The Brusselator in the unstable regime (A=1, B=3): The system approaches a limit cycle Bottom: The Brusselator in a stable regime with A=1 and B=1.7: For B2 the system is stable and approaches a fixed point.

physics function introduced by Eugen von Lommel

Technique for solving linear ordinary differential equations

ordinary differential equation

type of differential equation in mathematics

Empirical model for microorganisms growth

second order linear differential equation

On linear differential equations with certain properties

mathematical equation

computer modeling of time-varying behavior of a dynamical system

predicts density vs depth in Earth

equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation

mathematical function

Method for solving differential equations

Method for solving linear differential equations using the Laplace transform