Differential equation

File:Differential.png

An illustration of a differential equation. The arrows show how the differential equation locally influences a state, while the lines display how specific solutions are determined by starting conditions (red dots).

In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. Many of the fundamental laws of physics, chemistry, biology and economics can be formulated as differential equations. The question then becomes how to find the solutions of those equations.

The mathematical theory of differential equations has developed together with the sciences where the equations originate and where the results find application. Diverse scientific fields often give rise to identical problems in differential equations. In such cases, the mathematical theory can unify otherwise quite distinct scientific fields. A celebrated example is Fourier's theory of the conduction of heat in terms of sums of trigonometric functions, Fourier series, which finds application in the propagation of sound, the propagation of electric and magnetic fields, radio waves, optics, elasticity, spectral analysis of radiation, and other scientific fields.

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.

Mathematicians typically also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as shocks in hyperbolic (or wave) equations.

Types of differential equations

• 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.
• A delay differential equation (DDE) contains functions of one dependent variable, derivatives in that variable, and depends on previous states of the dependent variables.
• A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

Each of those categories is divided into linear and nonlinear subcategories. A differential equation is linear if it involves the unknown function and its derivatives only to the first power; otherwise the differential equation is nonlinear. Thus if $u'$ denotes the first derivative of u, then the equation

$u'=u$

is linear. while the equation

$u'=u^{2}$

is nonlinear. Solutions of a linear equation in which the unknown function or its derivative or derivatives appear in each term (linear homogeneous equations) may be added together or multiplied by an arbitrary constant in order to obtain additional solutions of that equation, but there is no general way to obtain families of solutions of nonlinear equations, except when they exhibit symmetries; see symmetries and invariants. Linear equations frequently appear as approximations to nonlinear equations, and these approximations are only valid under restricted conditions.

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.

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 emphasize differential equations from applications, and in addition to existence/uniqueness questions, are also concerned with rigorously justifying methods for approximating solutions. Physicists and engineers are usually more interested in computing approximate solutions to differential equations, and are typically less interested in justifications for whether these approximations really are close to the actual solutions. 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.

The study of the stability of solutions of differential equations is known as stability theory.

Famous differential equations

• Newton's Second Law in dynamics (mechanics)
• 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
• The geodesic 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

See also

• Picard–Lindelöf theorem on existence and uniqueness of solutions

References

• 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
• Wikibooks, Differential Equations
• E.L. Ince, Ordinary Differential Equations, Dover Publications, 1956

External links

• lectures on differential equations MIT Open CourseWare video
• Online Notes / Differential Equations Paul Dawkins, Lamar University
• Differential Equations, S.O.S. Mathematics
• Introduction to modeling via differential equations Introduction to modeling by means of differential equations, with critical remarks.
• Differential Equation Solver Java applet tool used to solve differential equations.