Differential equations are mathematical equations that involve one or more derivatives of an unknown function. They describe how a function's rate of change is related to the function itself. Differential equations are widely used in various fields, including physics, engineering, biology, economics, and many other scientific disciplines, to model dynamic systems and phenomena.

A general form of a differential equation involving a single variable y and its derivatives is given by:

F(x, y, y', y'', ..., y^(n)) = 0

Here, y' represents the first derivative of y with respect to x, y'' is the second derivative, and y^(n) is the n-th derivative. The order of the differential equation is determined by the highest-order derivative present.

Differential equations can be classified into various types based on their properties and solutions. Some common types include:

Ordinary Differential Equations (ODEs): Involving a single independent variable and its derivatives with respect to that variable.

Partial Differential Equations (PDEs): Involving multiple independent variables and their partial derivatives.

Linear Differential Equations: The unknown function and its derivatives appear linearly (not raised to a power or multiplied together).

Nonlinear Differential Equations: The unknown function or its derivatives appear nonlinearly.

Solving differential equations involves finding the function that satisfies the equation. This process may include finding an explicit formula for the solution or determining the behavior of the solution without providing an explicit formula.

Differential equations play a crucial role in modeling real-world phenomena, such as population dynamics, heat transfer, fluid flow, electrical circuits, and more. They provide a powerful tool for understanding and predicting the behavior of dynamic systems.