Differential equation
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. Differential equations have many applications in physics and chemistry, and are widespread in mathematical models explaining biological, social, and economic phenomena.
Differential equations are divided into two types:
- An ordinary differential equation (ODE) only contains functions of one variable, and derivatives in that variable.
- A partial differential equation (PDE) contains multivariate functions and their partial derivatives.
The order of a differential equation is that of the highest derivative that it contains. For instance, a first-order differential equation contains only first derivatives.
The study of differential equations is a wide field in both pure and applied mathematics. Pure mathematicians study the types and properties of differential equations, such as whether or not solutions exist, and should solutions exist, whether those solutions are unique. Applied mathematicians, physicists and engineers are usually more interested in how to compute solutions to differential equations. These solutions are then used to design bridges, automobiles, aircraft, sewers, etc. Often, these equations don't have closed form solutions and are solved using numerical methods.
Famous differential equations include:
- Maxwell's equations in electromagnetism
- The Schrödinger equation in quantum mechanics
- The heat equation in thermodynamics
- The Navier-Stokes equations in fluid dynamics
- The Lotka-Volterra equation in population dynamics
External links
- Polyanin, Andrei: EqWorld: The World of Mathematical Equations. An online resource focusing on ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
References
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, 2003.
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004.
- A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002.
- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.