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Ordinary differential equations, often abbreviated as ODEs, are mathematical equations that relate a function to its derivatives. In simpler terms, they describe how a quantity changes with respect to a single independent variable. This is in contrast to partial differential equations (PDEs), which involve functions of multiple independent variables and their partial derivatives.

The general form of an ordinary differential equation involves a function, say $y(x)$, and its derivatives with respect to $x$, such as $y'(x)$, $y''(x)$, and so on. The order of an ODE is determined by the highest order derivative present in the equation. For example, $y' = xy$ is a first-order ODE, while $y'' + y = 0$ is a second-order ODE.

ODEs are fundamental tools in many scientific and engineering disciplines. They are used to model a wide variety of phenomena, including the motion of objects, the spread of diseases, the flow of heat, and the behavior of electrical circuits. The solutions to ODEs are functions that satisfy the equation. Finding these solutions can range from straightforward analytical methods for simple equations to complex numerical techniques for more challenging ones. The study of ODEs involves understanding their properties, developing methods for solving them, and interpreting their solutions in the context of the problem being modeled.