A differential equation is a mathematical equation that relates a function with one or more of its derivatives. It is used to describe how a system changes over time based on its current state and rate of change.
Differential equations are important because they can be used to model and solve many real-world problems in various fields such as physics, engineering, economics, and biology. They also provide a way to analyze and understand the behavior of complex systems.
There are many techniques for solving differential equations, including separation of variables, substitution, and using a integrating factor. The method used depends on the type of differential equation and its complexity.
Ordinary differential equations (ODEs) involve a single independent variable, while partial differential equations (PDEs) involve multiple independent variables. ODEs are commonly used to model systems evolving in time, while PDEs are used to model systems evolving in multiple spatial dimensions.
Differential equations are used in many areas of science and engineering, including mechanics, electricity and magnetism, thermodynamics, and fluid dynamics. They are also used in economics and finance to model population growth, stock prices, and interest rates.