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Differential equation

From Wikipedia, the free encyclopedia
Type of functional equation (mathematics)
Not to be confused withDifference equation.
Differential equations
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Classification
Solution
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Inmathematics, adifferential equation is anequation that relates one or more unknownfunctions and theirderivatives.[1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common inmathematical models andscientific laws; therefore, differential equations play a prominent role in many disciplines includingengineering,physics,economics, andbiology.

The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Often when aclosed-form expression for the solutions is not available, solutions may be approximated numerically using computers, and manynumerical methods have been developed to determine solutions with a given degree of accuracy. Thetheory of dynamical systems analyzes thequalitative aspects of solutions, such as theiraverage behavior over a long time interval.

History

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Differential equations came into existence with theinvention of calculus byIsaac Newton andGottfried Leibniz. In Chapter 2 of his 1671 workMethodus fluxionum et Serierum Infinitarum,[2] Newton listed three kinds of differential equations:

dydx=f(x)dydx=f(x,y)x1∂y∂x1+x2∂y∂x2=y{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\[4pt]{\frac {dy}{dx}}&=f(x,y)\\[4pt]x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}}

In all these cases,y is an unknown function ofx (or ofx1 andx2), andf is a given function. He solved these examples and others using infinite series and discussed the non-uniqueness of solutions.

Jacob Bernoulli proposed theBernoulli differential equation in 1695.[3] This is anordinary differential equation of the form

y′+P(x)y=Q(x)yn{\displaystyle y'+P(x)y=Q(x)y^{n}\,}

for which the following year Leibniz obtained solutions by simplifying it.[4]

Historically, the problem of a vibrating string such as that of amusical instrument was studied byJean le Rond d'Alembert,Leonhard Euler,Daniel Bernoulli, andJoseph-Louis Lagrange.[5][6][7][8] In 1746, d’Alembert discovered the one-dimensionalwave equation, and within ten years Euler discovered the three-dimensional wave equation.[9]

TheEuler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of thetautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it tomechanics, which led to the formulation ofLagrangian mechanics.

In 1822,Fourier published his work onheat flow inThéorie analytique de la chaleur (The Analytic Theory of Heat),[10] in which he based his reasoning onNewton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of hisheat equation for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.

Example

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Inclassical mechanics, the motion of a body is described by its position and velocity as the time value varies.Newton's laws allow these variables to be expressed dynamically (given the position, velocity, and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In some cases, this differential equation (called anequation of motion) may be solved explicitly.

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

Types

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Differential equations can be classified several different ways. Besides describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Ordinary differential equations

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Main article:Ordinary differential equation

Anordinary differential equation (ODE) is an equation containing an unknownfunction of one real or complex variablex, its derivatives, and some given functions ofx. The unknown function is generally represented by adependent variable (often denotedy), which, therefore,depends onx. Thusx is often called theindependent variable of the equation. The term "ordinary" is used in contrast with the termpartial differential equation, which may be with respect tomore than one independent variable.

As, in general, the solutions of a differential equation cannot be expressed by aclosed-form expression,numerical methods are commonly used for solving differential equations on a computer.

Partial differential equations

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Main article:Partial differential equation

Apartial differential equation (PDE) is a differential equation that contains unknownmultivariable functions and theirpartial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or using a relevantcomputer model.

PDEs can be used to describe a wide variety of phenomena in nature such assound,heat,electrostatics,electrodynamics,fluid flow,elasticity, orquantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensionaldynamical systems, partial differential equations often modelmultidimensional systems.Stochastic partial differential equations generalize partial differential equations for modelingrandomness.

Linear differential equations

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Main article:linear differential equations

Linear differential equations are differential equations that arelinear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms ofintegrals.

Many differential equations that are encountered inphysics are linear, for example ODEs describingradioactive decay and PDEs forheat transfer by thermal diffusion. These lead tospecial functions, which may be defined as solutions of linear differential equations (seeHolonomic function).

Non-linear differential equations

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Main article:Non-linear differential equations

Anon-linear differential equation is a differential equation that is not alinear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particularsymmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic ofchaos. Even the fundamental questions of existence and uniqueness of solutions for nonlinear differential equations are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf.Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.[11]

In some circumstances, nonlinear differential equations may be approximated by linear ones. Theseapproximations are only valid under restricted conditions. For example, theharmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.Similarly, when a fixed point or stationary solution of a nonlinear differential equation has been found, investigation of its stability leads to a linear differential equation.

Equation order and degree

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Theorder of the differential equation is the highestorder of derivative of the unknown function that appears in the differential equation. For example, an equation containing onlyfirst-order derivatives is afirst-order differential equation, an equation containing thesecond-order derivative is asecond-order differential equation, and so on.[12][13]

When it is written as apolynomial equation in the unknown function and its derivatives, thedegree of the differential equation is, depending on the context, thepolynomial degree in the highest derivative of the unknown function,[14] or itstotal degree in the unknown function and its derivatives. In particular, alinear differential equation has degree one for both meanings, but the non-linear differential equationy′+y2=0{\displaystyle y'+y^{2}=0} is of degree one for the first meaning but not for the second one.

Differential equations that describe natural phenomena usually have only first and second order derivatives in them, but there are some exceptions, such as thethin-film equation, which is a fourth order partial differential equation.

Homogeneous linear equations

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A linear differential equation ishomogeneous if each term in the equation includes either the dependent variable or one of its derivatives. If this is not the case, so that there is a term that does not include either the dependent variable itself or a derivative of it, the equation isinhomogeneous orheterogeneous. See the examples section below.

Examples

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The first group of examples are ordinary differential equations, whereu is an unknown function ofx, andc andω are constants that are assumed to be known. These examples illustrate the distinction betweenlinear andnonlinear differential equations, and between homogeneous differential equations andinhomogeneous ones, defined above.

The next group of examples are partial differential equations. The unknown functionu depends on two variablesx andt orx andy.

Initial conditions and boundary conditions

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The general solution of a first-order ordinary differential equation includes a constant, which can be thought of as a constant of integration. Similarly, the general solution of an{\displaystyle n}th order ODE containsn{\displaystyle n} constants.

To determine the values of these constants, additional conditions must be provided. If the independent variable corresponds to time, this information takes the form ofinitial conditions. For example, for a second-order ODE describing the motion of a particle, the initial conditions would typically be the position and velocity of the particle at the initial time. The ODE and its initial conditions form what is known as aninitial value problem.

For the case of a spatial independent variable, these conditions are generally known as boundary conditions. These are often specified at different values of the independent variable. Examples include the motion of a vibrating string that is fixed at two endpoints. In this case the ODE and boundary conditions lead to aboundary value problem.

More generally, the terminitial conditions is normally used when the conditions are given at the same value of the independent variable, and the termboundary conditions is used when they are specified at different values of the independent variable. In either case, the number of initial or boundary conditions should match the order of the differential equation.

Existence of solutions

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For a given differential equation, the questions of whether solutions are unique or exist at all are notable subjects of interest.

For a first-order initial value problem, thePeano existence theorem gives one set of circumstances in which a solution exists. Given any point(a,b){\displaystyle (a,b)} in the xy-plane, define some rectangular regionZ{\displaystyle Z}, such thatZ=[l,m]×[n,p]{\displaystyle Z=[l,m]\times [n,p]} and(a,b){\displaystyle (a,b)} is in the interior ofZ{\displaystyle Z}. If we are given a differential equationdydx=g(x,y){\textstyle {\frac {dy}{dx}}=g(x,y)} and the condition thaty=b{\displaystyle y=b} whenx=a{\displaystyle x=a}, then there is locally a solution to this problem ifg(x,y){\displaystyle g(x,y)} is continuous onZ{\displaystyle Z}. This solution exists on some interval with its center ata{\displaystyle a}. The solution may not be unique. (SeeOrdinary differential equation for other results.)

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

fn(x)dnydxn+⋯+f1(x)dydx+f0(x)y=g(x){\displaystyle f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)}

such that

y(x0)=y0,y′(x0)=y0′,y″(x0)=y0″,…{\displaystyle {\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}}

For any nonzerofn(x){\displaystyle f_{n}(x)}, if{f0,f1,…}{\displaystyle \{f_{0},f_{1},\ldots \}} andg{\displaystyle g} are continuous on some interval containingx0{\displaystyle x_{0}},y{\displaystyle y} exists and is unique.[15]

Related concepts

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Connection to difference equations

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Differential equations are closely related todifference equations, in which the independent variable assumes only discrete values, and the equation relates the value of the unknown function at a point to its values at nearby points. Many numerical methods for differential equations, for example theEuler method, involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Applications

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The study of differential equations is a wide field inpure andapplied mathematics,physics, andengineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics is concerned with finding solutions, either directly or approximately, and studying their behaviour. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not haveclosed form solutions. Instead, solutions can be approximated usingnumerical methods.

Many fundamental laws ofphysics andchemistry can be formulated as differential equations. Inbiology andeconomics, differential equations are used tomodel the behavior ofcomplex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. When this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-orderpartial differential equation, thewave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed byJoseph Fourier, is governed by another second-order partial differential equation, theheat equation. It turns out that manydiffusion processes, while seemingly different, are described by the same equation; theBlack–Scholes equation in finance is, for instance, related to the heat equation.

The number of differential equations that have received a name, in various scientific areas, demonstrates the importance of the topic. SeeList of named differential equations.

Software

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SomeCAS software can solve differential equations. These are the commands used in the leading programs:

See also

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References

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  1. ^Dennis G. Zill (15 March 2012).A First Course in Differential Equations with Modeling Applications. Cengage Learning.ISBN 978-1-285-40110-2.
  2. ^Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].
  3. ^Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis",Acta Eruditorum
  4. ^Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993),Solving ordinary differential equations I: Nonstiff problems, Berlin, New York:Springer-Verlag,ISBN 978-3-540-56670-0
  5. ^Frasier, Craig (July 1983)."Review ofThe evolution of dynamics, vibration theory from 1687 to 1742, by John T. Cannon and Sigalia Dostrovsky"(PDF).Bulletin of the American Mathematical Society. New Series.9 (1).
  6. ^Wheeler, Gerard F.; Crummett, William P. (1987). "The Vibrating String Controversy".Am. J. Phys.55 (1):33–37.Bibcode:1987AmJPh..55...33W.doi:10.1119/1.15311.
  7. ^For a special collection of the 9 groundbreaking papers by the three authors, seeFirst Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating stringsArchived 2020-02-09 at theWayback Machine (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
  8. ^For Lagrange's contributions to the acoustic wave equation, consultAcoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
  9. ^Speiser, David.Discovering the Principles of Mechanics 1600-1800, p. 191 (Basel: Birkhäuser, 2008).
  10. ^Fourier, Joseph (1822).Théorie analytique de la chaleur (in French). Paris: Firmin Didot Père et Fils.OCLC 2688081.
  11. ^Boyce, William E.; DiPrima, Richard C. (1967).Elementary Differential Equations and Boundary Value Problems (4th ed.). John Wiley & Sons. p. 3.
  12. ^Weisstein, Eric W. "Ordinary Differential Equation Order." FromMathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
  13. ^Order and degree of a differential equation, accessed Dec 2015.
  14. ^Elias Loomis (1887).Elements of the Differential and Integral Calculus (revised ed.). Harper & Bros. p. 247.Extract of page 247
  15. ^Zill, Dennis G. (2001).A First Course in Differential Equations (5th ed.). Brooks/Cole.ISBN 0-534-37388-7.
  16. ^"dsolve - Maple Programming Help".www.maplesoft.com. Retrieved2020-05-09.
  17. ^"DSolve - Wolfram Language Documentation".www.wolfram.com. Retrieved2020-06-28.
  18. ^Schelter, William F. Gaertner, Boris (ed.)."Differential Equations - Symbolic Solutions".The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation onSourceForge).Archived from the original on 2022-10-04.
  19. ^"Basic Algebra and Calculus — Sage Tutorial v9.0".doc.sagemath.org. Retrieved2020-05-09.
  20. ^"ODE".SymPy 1.11 documentation. 2022-08-22.Archived from the original on 2022-09-26.
  21. ^"Symbolic algebra and Mathematics with Xcas"(PDF).

Further reading

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External links

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Wikiquote has quotations related toDifferential equation.
Wikibooks has a book on the topic of:Ordinary Differential Equations
Wikiversity has learning resources aboutDifferential equations
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