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Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental tools in various scientific and engineering disciplines for modeling and understanding dynamic systems. The order of a differential equation is determined by the highest derivative present in the equation.

There are two main types of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single independent variable and their derivatives. For instance, Newton's second law of motion, when expressed in terms of velocity or position as a function of time, is an ODE. PDEs, on the other hand, involve functions of multiple independent variables and their partial derivatives. These are used to model phenomena that vary across space and time, such as heat diffusion or wave propagation.

The solution to a differential equation is a function that satisfies the equation. Finding these solutions can be challenging, and various analytical and numerical methods exist. Analytical methods provide exact solutions, while numerical methods approximate solutions, especially for complex equations where analytical solutions are not feasible. The study and application of differential equations are crucial for advancements in physics, engineering, biology, economics, and many other fields.