Differential equations are mathematical equations that involve an unknown function and its derivatives. They are used to model relationships between various quantities and their rates of change. Differential equations are commonly used in physics, engineering, biology, economics, and many other fields to describe the behavior of complex systems. There are two main types of differential equations: ordinary differential equations (ODEs), which involve one independent variable, and partial differential equations (PDEs), which involve multiple independent variables. Solutions to differential equations can be found analytically or numerically using various methods such as separation of variables, series solutions, and numerical approximation techniques. Overall, the study of differential equations is crucial for understanding and modeling dynamic systems and phenomena in the natural and social sciences.