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 independent variable, and derivatives in that variable.
- A partial differential equation (PDE) contains functions of multiple independent variables 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 they exist, whether they 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 simulate celestial motions, design bridges, automobiles, aircraft, sewers, etc. Often, these equations do not have closed form solutions and are solved using numerical methods.
Famous differential equations include:
- Maxwell's equations in electromagnetism
- Einstein's field equation in general relativity
- The Schrödinger equation in quantum mechanics
- The heat equation in thermodynamics
- The wave equation
- Laplace's equation, which defines harmonic functions
- Poisson's equation
- The Navier-Stokes equations in fluid dynamics
- The Lotka-Volterra equation in population dynamics
- The Black-Scholes equation in finance
- The Cauchy-Riemann equations in complex analysis
- D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.
- W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
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