differentialequation

A differential equation is a mathematical equation that relates a function with one or more of its derivatives. Differential equations are used to describe a broad range of phenomena in science and engineering, as they can model the rate at which a quantity changes. The unknown function is typically denoted by a symbol like y, and its derivatives are represented using prime notation (y', y'') or Leibniz notation (dy/dx, d^2y/dx^2).

The order of a differential equation is determined by the highest order derivative that appears in the equation. For example, y' + 2y = x is a first-order differential equation, while y'' - 3y' + y = sin(x) is a second-order differential equation. Differential equations can also be classified as linear or nonlinear. A linear differential equation is one where the unknown function and its derivatives appear only in a linear fashion, meaning they are not multiplied by each other or by the function itself.

Solving a differential equation means finding a function that, when substituted into the equation, makes the equation true. The solutions to differential equations can be found using various analytical and numerical methods. Analytical methods aim to find an exact symbolic expression for the solution, while numerical methods provide approximate solutions, often used when analytical solutions are difficult or impossible to obtain. The study of differential equations is a fundamental part of calculus and has wide-ranging applications in physics, chemistry, biology, economics, and many other fields.