Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model various physical phenomena and are an important tool in many scientific fields.
A solution of a differential equation is a function that satisfies the equation. In other words, when the solution is plugged into the equation, it will make the equation true.
There are various methods for solving differential equations, including separation of variables, substitution, and using specific formulas for certain types of equations. Some equations may also require numerical methods to find an approximate solution.
Initial conditions are values given to the function and its derivatives at a specific point. These conditions are necessary to find a unique solution for a differential equation. They can represent the starting point or some other known information about the system being modeled.
Differential equations are used to model a wide range of physical phenomena, including growth and decay, motion, heat transfer, and population dynamics. They are also used in fields such as physics, engineering, economics, and biology to analyze and predict the behavior of systems.