In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below). The derivatives are ordinary because partial derivatives only apply to functions of many independent variables (see Partial differential equation).
The subject of ODEs is a sophisticated one (more so with PDEs), primarily due to the various forms the ODE can take and how they can be integrated. Linear differential equations are ones with solutions that can be added and multiplied by coefficients, and the theory of linear differential equations is well-defined and understood, and exact closed form solutions can be obtained. By contrast, ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary functions in closed form — rather the exact (or "analytic") solutions are in series or integral form. Frequently graphical and numerical methods are used to generate solutions, by hand or on computer (only approximately, but possible to do very accurately depending on the specific method used), because in this way the properties of the solutions without solving them can still yield very useful information, which may be all that is needed.
Read more about Ordinary Differential Equation: Background, Definitions, Existence and Uniqueness of Solutions, Reduction of Order, Summary of Exact Solutions, Software For ODE Solving
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