A differential equation is a mathematical equation that relates the rate of change of a function to the function itself. It involves derivatives, which represent the rate of change, and the function itself.
The specific method for solving a differential equation depends on the type of equation. Some common techniques include separation of variables, substitution, and integrating factors. It is also important to identify any initial conditions or boundary conditions that may be given in the problem.
The order of a differential equation is the highest derivative present in the equation. For example, a first-order differential equation has only one derivative, while a second-order differential equation has a second derivative.
An ordinary differential equation involves only one independent variable, while a partial differential equation involves more than one independent variable. This means that the derivatives in a partial differential equation can be taken with respect to different variables, while the derivatives in an ordinary differential equation are all taken with respect to the same variable.
Differential equations are used to model and describe many physical phenomena in science and engineering, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits. They also have applications in economics, biology, and other fields. By solving differential equations, scientists and engineers can make predictions and analyze systems in order to better understand the world around us.