Differential equation

A differential equation is a mathematical equation that involves variables like  or , as well as the rate at which those variables change. Differential equations are special because the solution of a differential equation is itself a function instead of a number.

In applications of mathematics, problems often arise in which the dependence of one parameter on another is unknown, but it is possible to write an expression for the rate of change of one parameter relative to another (derivative). In this case, the problem is reduced to finding a function by its derivative related to some other expressions.

Types of differential equations
If a differential equation only involves  and its derivative, the rate at which  changes, then it is called a first order differential equation. A higher-order differential equation has derivatives of other derivatives. If there are more variables than just  and , then it is said to be a partial differential equation. Sometimes, something will obey several differential equations at the same time. These are said to be modeled by coupled differential equations.

Some differential equations can be solved exactly, and some cannot. Sometimes one can only be estimated, and a computer program can do this very fast. Although they may seem overly-complicated to someone who has not studied differential equations before, the people who use differential equations tell us that they would not be able to figure important things out without them. Most scientists and engineers (as well as mathematicians) take at least one course in differential equations while in college. Some mathematicians devote their career to investigating differential equations that are difficult to solve.

 Uses 
Differential equations are used in many fields of science since they describe real-world things:
 In physics for various forms of movement, or oscillations
 Radioactive decay is calculated using differential equations.
 Isaac Newton's Second law of motion 
 Newton's Law of Cooling
 The wave equation
 Laplace's equation
 The Navier–Stokes equations described the movement of fluids
 The Hamiltonian equations for general mechanics

People who studied about differential equations

 Carl Gustav Jacob Jacobi
 Hiroshi Umemura
 Israel Gelfand
 Peter Lax
 Ryogo Hirota
 Sofya Kovalevskaya
 Vladimir Arnold

Related pages
 Finite element method
 Numerical methods for ordinary differential equations
 Numerical methods for partial differential equations

Category:Differential equations