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Ordinary differential equation - Definition | | Ordinary differential equation - Definition: Encyclopedia II - Ordinary differential equation - Definition | | Let y represent an unknown function of x, and let
denote the derivatives
An ordinary differential equation (ODE) is an equation involving
The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of ...
See also:Ordinary differential equation, Ordinary differential equation - Definition, Ordinary differential equation - General application, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - Types of differential equations with some history, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Linear PDEs, Ordinary differential equation - First-order PDEs, Ordinary differential equation - Singular solutions, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Lie's theory, Ordinary differential equation - Bibliography | | | Ordinary differential equation, Ordinary differential equation - Bibliography, Ordinary differential equation - Definition, Ordinary differential equation - Existence and nature of solutions, Ordinary differential equation - First-order PDEs, Ordinary differential equation - General application, Ordinary differential equation - General solution method for first-order linear ODEs, Ordinary differential equation - Homogeneous linear ODEs with constant coefficients, Ordinary differential equation - Lie's theory, Ordinary differential equation - Linear ODEs with constant coefficients, Ordinary differential equation - Linear ODEs with variable coefficient, Ordinary differential equation - Linear PDEs, Ordinary differential equation - Reduction to quadratures, Ordinary differential equation - Singular solutions, Ordinary differential equation - The Fuchsian theory, Ordinary differential equation - Types of differential equations with some history, Examples of differential equations, Differential equations of mathematical physics, Differential equations from outside physics, Difference equation, Laplace transform applied to differential equations, Boundary value problem, List of dynamical systems and differential equations topics | | |
| | | Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Definition
Ordinary differential equation - Definition
Let y represent an unknown function of x, and let
denote the derivatives
An ordinary differential equation (ODE) is an equation involving
The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of the equation is the highest power of the highest derivative.
A solution of an ODE is a function y(x) whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique. A general solution of an nth-order equation is a solution containing n arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. A singular solution is a solution that can't be derived from the general solution.
When a differential equation of order n has the form
it is called an implicit differential equation whereas the form
is called an explicit differential equation.
A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.
Other related archivesAbelian integrals, Bernoullis, Boole, Boundary value problem, Cauchy, Cauchy-Kovalevskaya theorem, Clairaut, Clebsch, Cramer's rule, Darboux, Difference equation, Differential equations from outside physics, Differential equations of mathematical physics, Euler, Euler's formula, Examples of differential equations, Gauss, Im(y), Jacobi, Jean Bernoulli, Lagrange, Laplace transform applied to differential equations, Leibniz, Levy, Lie, Lie groups, Lie's, Lipschitz condition, List of dynamical systems and differential equations topics, Method of undetermined coefficients, Method of variation of parameters, Monge, Newton, Poincaré-Bendixson theorem, Re(y), Riccati, Solving, Wronskian, analysis, analytic function, basis, boundary conditions, complex numbers, conjugate, constants of integration, convergent series, curve, d'Alembert, derivatives, differential calculus, differential equation, differential operator, discriminant, equivalence classes, harmonic oscillator, iff, infinitesimal transformations, integral calculus, linear, linear combination, linear combinations, list of integrals of exponential functions, mathematics, nineteenth century, nondimensionalization, numerical ordinary differential equations, partial derivatives, partial differential equations, quadratic equation, quadratures, singular solution, singular solutions, symplectic topology, total differential equations, variation of parameters, vector fields, zero
 Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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