Differential equation

A differential equation is a mathematical equation that involves variables like $x$ or $y$, 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.

A picture of airflow, modeled using a differential equation.

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 $x$ and its derivative, the rate at which $x$ 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 $x$ and $y$, 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