3414:. 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 emphasizes the rigorous justification of the methods for approximating solutions. 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 necessarily be directly solvable, i.e. do not have
4896:
4908:
3390:, 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 the approximation of the solution of a differential equation by the solution of a corresponding difference equation.
4932:
1243:
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.
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to identical differential equations. Whenever 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-order
1481:. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf.
3117:
748:. 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; therefore, differential equations play a prominent role in many disciplines including
3445:
the behavior of complex 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
1242:
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
767:
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
1251:
Differential equations can be divided into several types. Apart from 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
1183:
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 to
1472:
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 particular
2366:
819:
3661:(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",
2250:
2944:
2146:
1231:
allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.
1399:, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant
824:
1812:
2955:
1252:
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.
1895:
2265:
2046:
2960:
1013:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=f(x)\\{\frac {dy}{dx}}&=f(x,y)\\x_{1}{\frac {\partial y}{\partial x_{1}}}&+x_{2}{\frac {\partial y}{\partial x_{2}}}=y\end{aligned}}}
1211:, 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 his
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Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the
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456:
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3454:, 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 by
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1434:. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional
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are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between
1485:). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.
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The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See
261:
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in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of
708:
163:
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582:
236:
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4349:
3112:{\displaystyle {\begin{aligned}y(x_{0})&=y_{0},&y'(x_{0})&=y'_{0},&y''(x_{0})&=y''_{0},&\ldots \end{aligned}}}
2385:. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.
3648:
Newton, Isaac. (c.1671). Methodus
Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 .
1728:
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may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains
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First
Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings
3585:
2257:
1827:
1215:
for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.
636:
361:
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5010:
4912:
3597:
3294:, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
2775:
2361:{\displaystyle {\frac {\partial u}{\partial t}}=6u{\frac {\partial u}{\partial x}}-{\frac {\partial ^{3}u}{\partial x^{3}}}.}
1271:
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1976:
1477:. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of
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656:
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equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.
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of the unknown function that appears in the differential equation. For example, an equation containing only
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to nonlinear equations. These approximations are only valid under restricted conditions. For example, the
1388:
1358:
1290:
1076:
772:
741:
400:
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2152:
Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the
1056:
He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.
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3338:
3333:(SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in
2782:
1963:
Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a
1165:
646:
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520:
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3371:(DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
3334:
1427:
1310:
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532:
468:
441:
2245:{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}
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Homogeneous second-order linear constant coefficient ordinary differential equation describing the
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1224:
796:
775:
for the solutions is not available, solutions may be approximated numerically using computers. The
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808:
559:
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software can solve differential equations. These are the commands used in the leading programs:
3467:
2939:{\displaystyle f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)}
1558:
3311:(IDE) is an equation that combines aspects of a differential equation and an integral equation.
2447:
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681:
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1227:, the motion of a body is described by its position and velocity as the time value varies.
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64:
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was developed in the 1750s by Euler and
Lagrange in connection with their studies of the
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1655:
Heterogeneous first-order linear constant coefficient ordinary differential equation:
5704:
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5591:
4635:
4567:
4519:
4190:
4186:
4123:
3990:
3735:
Wheeler, Gerard F.; Crummett, William P. (1987). "The
Vibrating String Controversy".
3459:
3451:
2141:{\displaystyle {\frac {\partial u}{\partial t}}+t{\frac {\partial u}{\partial x}}=0.}
1489:
1212:
1169:
4266:
5451:
5386:
5300:
4577:
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4284:
Introduction to modeling by means of differential equations, with critical remarks.
1548:
1478:
800:
661:
611:
497:
125:
3820:
3770:
For a special collection of the 9 groundbreaking papers by the three authors, see
1357:
As, in general, the solutions of a differential equation cannot be expressed by a
780:
3898:
3356:. Mathematical models that involve ultrametric pseudo-differential equations use
5208:
4779:
4443:
4366:
4296:
4000:
3794:
Allan D. Pierce, Acoustical Soc of
America, 1989; page 18.(retrieved 9 Dec 2012)
3411:
1180:
749:
729:
69:
4301:
5521:
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4643:
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processes, while seemingly different, are described by the same equation; the
1447:
1423:
745:
686:
3999:
The
Computer Algebra Program Maxima - a Tutorial (in Maxima documentation on
1172:, and within ten years Euler discovered the three-dimensional wave equation.
5198:
4275:
3862:
3463:
3438:
3430:
3322:
and the equation involves some known stochastic processes, for example, the
1200:
1185:
787:
have been developed to determine solutions with a given degree of accuracy.
757:
427:
148:
91:
81:
3830:
3790:
For de
Lagrange's contributions to the acoustic wave equation, can consult
2392:
gives one set of circumstances in which a solution exists. Given any point
1555:
has degree one for both meanings, but the non-linear differential equation
1406:
PDEs can be used to describe a wide variety of phenomena in nature such as
1145:
for which the following year
Leibniz obtained solutions by simplifying it.
4334:
4246:
5678:
5516:
5511:
5238:
4948:
4668:
4587:
4514:
3520:
3458:, is governed by another second-order partial differential equation, the
3386:
The theory of differential equations is closely related to the theory of
3301:
3290:(DDE) is an equation for a function of a single variable, usually called
1964:
1474:
768:
differential equation may be determined without computing them exactly.
4453:
3434:
3426:
3407:
1343:
761:
753:
102:
97:
86:
3792:
Acoustics: An
Introduction to Its Physical Principles and Applications
2256:
Homogeneous third-order non-linear partial differential equation, the
4152:
3757:
3470:
equation in finance is, for instance, related to the heat equation.
1901:
Heterogeneous first-order nonlinear ordinary differential equation:
1365:
are commonly used for solving differential equations on a computer.
783:
analysis of systems described by differential equations, while many
4317:
An introductory textbook on differential equations by Jiri Lebl of
3867:
http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html
3625:
A First Course in Differential Equations with Modeling Applications
1599:
is of degree one for the first meaning but not for the second one.
3529:
2785:. Suppose we had a linear initial value problem of the nth order:
1407:
1350:
may be defined as solutions of linear differential equations (see
1207:(The Analytic Theory of Heat), in which he based his reasoning on
1148:
Historically, the problem of a vibrating string such as that of a
4326:
Topics covered in a first year course in differential equations.
4318:
4314:
3538:
1723:
Homogeneous second-order linear ordinary differential equation:
1411:
5471:
4952:
4338:
4160:
Handbook of Exact Solutions for Ordinary Differential Equations
1807:{\displaystyle {\frac {d^{2}u}{dx^{2}}}-x{\frac {du}{dx}}+u=0.}
4260:
2077:
Homogeneous first-order linear partial differential equation:
1551:
in the unknown function and its derivatives. In particular, a
4114:
Elementary Differential Equations and Boundary Value Problems
3846:
Elementary Differential Equations and Boundary Value Problems
3709:
The evolution of dynamics, vibration theory from 1687 to 1742
3677:
Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993),
3679:
Solving ordinary differential equations I: Nonstiff problems
2754:. This solution exists on some interval with its center at
4162:(2nd ed.). Boca Raton: Chapman & Hall/CRC Press.
1547:
in the highest derivative of the unknown function, or its
4146:
A Treatise on Ordinary and Partial Differential Equations
1606:, which is a fourth order partial differential equation.
3418:
solutions. Instead, solutions can be approximated using
3318:(SDE) is an equation in which the unknown quantity is a
1890:{\displaystyle {\frac {d^{2}u}{dx^{2}}}+\omega ^{2}u=0.}
4315:
Notes on Diffy Qs: Differential Equations for Engineers
3398:
The study of differential equations is a wide field in
1446:
generalize partial differential equations for modeling
813:, Newton listed three kinds of differential equations:
4324:
Khan Academy Video playlist on differential equations
2703:
2662:, then there is locally a solution to this problem if
2558:
4330:
MathDiscuss Video playlist on differential equations
4192:
Ordinary Differential Equations and Dynamical Systems
3576:
Numerical methods for ordinary differential equations
3263:
3236:
3216:
3164:
3128:
2958:
2794:
2760:
2740:
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2616:
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2430:
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2085:
1979:
1909:
1830:
1731:
1663:
1561:
1289:. The unknown function is generally represented by a
1235:
In some cases, this differential equation (called an
1168:. In 1746, d’Alembert discovered the one-dimensional
1079:
822:
4302:
Collection of ODE and DAE models of physical systems
4177:
Porter, R. I. (1978). "XIX Differential Equations".
3581:
Numerical methods for partial differential equations
2053:
In the next group of examples, the unknown function
2041:{\displaystyle L{\frac {d^{2}u}{dx^{2}}}+g\sin u=0.}
795:
Differential equations came into existence with the
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5153:
5115:
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4993:
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4452:
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4282:
Introduction to modeling via differential equations
3903:(revised ed.). Harper & Bros. p. 247.
1488:Linear differential equations frequently appear as
1387:) is a differential equation that contains unknown
4297:Exact Solutions of Ordinary Differential Equations
4122:
3900:Elements of the Differential and Integral Calculus
3269:
3249:
3222:
3202:
3150:
3111:
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1951:
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1711:
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1134:
1012:
5366:List of nonlinear ordinary differential equations
4153:University of Michigan Historical Math Collection
4021:"Basic Algebra and Calculus — Sage Tutorial v9.0"
3848:(4th ed.). John Wiley & Sons. p. 3.
3806:Discovering the Principles of Mechanics 1600-1800
3781:(retrieved 13 Nov 2012). Herman HJ Lynge and Son.
1539:in the unknown function and its derivatives, its
5371:List of nonlinear partial differential equations
3433:can be formulated as differential equations. In
2424:in the xy-plane, define some rectangular region
3861:. "Ordinary Differential Equation Order." From
3844:Boyce, William E.; DiPrima, Richard C. (1967).
3825:(in French). Paris: Firmin Didot Père et Fils.
1285:, its derivatives, and some given functions of
5361:List of linear ordinary differential equations
18:Differential equations of mathematical physics
5483:
4964:
4350:
3995:"Differential Equations - Symbolic Solutions"
3718:Bulletin of the American Mathematical Society
2781:However, this only helps us with first order
2727:{\textstyle {\frac {\partial g}{\partial x}}}
1438:, partial differential equations often model
709:
8:
4067:"Symbolic algebra and Mathematics with Xcas"
3197:
3165:
2388:For first order initial value problems, the
3878:Order and degree of a differential equation
3711:, by John T. Cannon and Sigalia Dostrovsky"
3230:are continuous on some interval containing
2381:differential equations is not like solving
5490:
5476:
5468:
5157:
4990:
4971:
4957:
4949:
4357:
4343:
4335:
2552:. If we are given a differential equation
1712:{\displaystyle {\frac {du}{dx}}=cu+x^{2}.}
810:Methodus fluxionum et Serierum Infinitarum
716:
702:
182:
57:
36:
4125:Theory of Ordinary Differential Equations
3967:"DSolve - Wolfram Language Documentation"
3262:
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2008:
1990:
1983:
1978:
1952:{\displaystyle {\frac {du}{dx}}=u^{2}+4.}
1937:
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1468:is a differential equation that is not a
1444:Stochastic partial differential equations
1131:
1125:
1078:
991:
973:
967:
947:
929:
923:
870:
827:
823:
821:
27:Type of functional equation (mathematics)
4158:Polyanin, A. D.; Zaitsev, V. F. (2003).
4121:Coddington, E. A.; Levinson, N. (1955).
3919:A First Course in Differential Equations
3588:on existence and uniqueness of solutions
3331:stochastic partial differential equation
1331:are the differential equations that are
1280:function of one real or complex variable
3614:
3203:{\displaystyle \{f_{0},f_{1},\ldots \}}
2774:. The solution may not be unique. (See
1278:) is an equation containing an unknown
597:
348:
270:
228:
171:
138:
110:
60:
46:
39:
1483:Navier–Stokes existence and smoothness
4267:Online Notes / Differential Equations
3594:, also known as 'difference equation'
3441:, differential equations are used to
7:
5356:List of named differential equations
4919:
3475:List of named differential equations
2603:{\textstyle {\frac {dy}{dx}}=g(x,y)}
1640:differential equations, and between
1317:" is used in contrast with the term
1135:{\displaystyle y'+P(x)y=Q(x)y^{n}\,}
164:List of named differential equations
5281:Method of undetermined coefficients
5062:Dependent and independent variables
4931:
3809:, p. 191 (Basel: Birkhäuser, 2008).
3326:in the case of diffusion equations.
1541:degree of the differential equation
237:Dependent and independent variables
4258:Lectures on Differential Equations
4221:Handbook of Differential Equations
3534:sympy.solvers.ode.dsolve(equation)
3376:Connection to difference equations
2715:
2707:
2339:
2325:
2309:
2301:
2280:
2272:
2220:
2206:
2183:
2169:
2123:
2115:
2097:
2089:
1543:is, depending on the context, the
1530:second-order differential equation
1506:order of the differential equation
1342:Most ODEs that are encountered in
1188:, which led to the formulation of
984:
976:
940:
932:
25:
4218:Daniel Zwillinger (12 May 2014).
3943:"dsolve - Maple Programming Help"
1521:first-order differential equation
1460:Non-linear differential equations
1454:Non-linear differential equations
740:that relates one or more unknown
5178:Carathéodory's existence theorem
4930:
4918:
4907:
4906:
4894:
4245:
4055:from the original on 2022-09-26.
4009:from the original on 2022-10-04.
3822:Théorie analytique de la chaleur
3622:Dennis G. Zill (15 March 2012).
3603:System of differential equations
3561:Functional differential equation
3316:stochastic differential equation
1466:non-linear differential equation
1205:Théorie analytique de la chaleur
807:. In Chapter 2 of his 1671 work
372:Carathéodory's existence theorem
5622:Least-squares spectral analysis
5549:Fundamental theorem of calculus
4815:Computational complexity theory
4136:Ordinary Differential Equations
3369:differential algebraic equation
1614:In the first group of examples
1397:ordinary differential equations
1321:, which may be with respect to
1256:Ordinary differential equations
1064:Bernoulli differential equation
4086:Abbott, P.; Neill, H. (2003).
3598:Abstract differential equation
3348:is an equation which contains
3145:
3139:
3074:
3061:
3025:
3012:
2979:
2966:
2933:
2927:
2915:
2909:
2873:
2867:
2811:
2805:
2776:Ordinary differential equation
2684:
2672:
2597:
2585:
2519:
2507:
2487:
2475:
2469:
2457:
2411:
2399:
1369:Partial differential equations
1272:ordinary differential equation
1262:Ordinary differential equation
1118:
1112:
1100:
1094:
1068:ordinary differential equation
912:
900:
863:
857:
459: / Integral solutions
1:
4288:Mathematical Assistant on Web
4201:American Mathematical Society
3921:(5th ed.). Brooks/Cole.
3448:partial differential equation
3358:pseudo-differential operators
3309:integro-differential equation
1524:, an equation containing the
1381:partial differential equation
1375:Partial differential equation
1329:Linear differential equations
1319:partial differential equation
5006:Notation for differentiation
3705:Frasier, Craig (July 1983).
3346:pseudo-differential equation
1553:linear differential equation
1346:are linear. Therefore, most
1266:Linear differential equation
1239:) may be solved explicitly.
503:Exponential response formula
249:Coupled / Decoupled
5102:Exact differential equation
4179:Further Elementary Analysis
3556:Exact differential equation
3288:delay differential equation
1313:of the equation. The term "
777:theory of dynamical systems
5727:
4865:Films about mathematicians
3865:--A Wolfram Web Resource.
3379:
1618:is an unknown function of
1592:{\displaystyle y'+y^{2}=0}
1457:
1395:. (This is in contrast to
1372:
1325:one independent variable.
1259:
1027:is an unknown function of
29:
5687:
5587:
5506:
5412:Józef Maria Hoene-Wroński
5392:Gottfried Wilhelm Leibniz
5183:Cauchy–Kowalevski theorem
4888:
4434:Philosophy of mathematics
4374:
4290:Symbolic ODE tool, using
3462:. It turns out that many
3425:Many fundamental laws of
2493:{\displaystyle Z=\times }
2057:depends on two variables
1500:Equation order and degree
637:Józef Maria Hoene-Wroński
583:Undetermined coefficients
492:Method of characteristics
377:Cauchy–Kowalevski theorem
5306:Finite difference method
4870:Recreational mathematics
4049:SymPy 1.11 documentation
3917:Zill, Dennis G. (2001).
3819:Fourier, Joseph (1822).
3151:{\displaystyle f_{n}(x)}
1535:When it is written as a
1440:multidimensional systems
362:Picard–Lindelöf theorem
356:Existence and uniqueness
30:Not to be confused with
5286:Variation of parameters
5276:Separation of variables
5173:Peano existence theorem
5168:Picard–Lindelöf theorem
5055:Attributes of variables
4755:Mathematical statistics
4745:Mathematical psychology
4715:Engineering mathematics
4649:Algebraic number theory
4088:Teach Yourself Calculus
3993:Gaertner, Boris (ed.).
3586:Picard–Lindelöf theorem
2734:are both continuous on
2610:and the condition that
2390:Peano existence theorem
1526:second-order derivative
1516:first-order derivatives
1389:multivariable functions
1209:Newton's law of cooling
1177:Euler–Lagrange equation
1154:Jean le Rond d'Alembert
588:Variation of parameters
578:Separation of variables
367:Peano existence theorem
5711:Differential equations
5554:Calculus of variations
5527:Differential equations
5447:Carl David Tolmé Runge
5021:Differential-algebraic
4980:Differential equations
4901:Mathematics portal
4750:Mathematical sociology
4730:Mathematical economics
4725:Mathematical chemistry
4654:Analytic number theory
4535:Differential equations
4276:Differential Equations
4263:Open CourseWare Videos
4252:Differential equations
4149:. John Wiley and Sons.
4101:Differential Equations
4099:; Hall, G. R. (2006).
3362:differential operators
3277:exists and is unique.
3271:
3251:
3224:
3204:
3152:
3113:
2940:
2783:initial value problems
2768:
2748:
2728:
2691:
2690:{\displaystyle g(x,y)}
2656:
2630:
2604:
2546:
2532:is in the interior of
2526:
2494:
2438:
2418:
2374:Existence of solutions
2362:
2246:
2142:
2042:
1953:
1891:
1808:
1713:
1645:differential equations
1593:
1359:closed-form expression
1199:published his work on
1136:
1014:
773:closed-form expression
657:Carl David Tolmé Runge
200:Differential-algebraic
41:Differential equations
5647:Representation theory
5606:quaternionic analysis
5602:Hypercomplex analysis
5500:mathematical analysis
5432:Augustin-Louis Cauchy
5427:Joseph-Louis Lagrange
5321:Finite element method
5311:Crank–Nicolson method
5245:Numerical integration
5224:Exponential stability
5116:Relation to processes
5001:Differential operator
4880:Mathematics education
4810:Theory of computation
4530:Hypercomplex analysis
3897:Elias Loomis (1887).
3339:statistical mechanics
3272:
3252:
3250:{\displaystyle x_{0}}
3225:
3205:
3153:
3114:
2941:
2769:
2749:
2729:
2692:
2657:
2631:
2605:
2547:
2527:
2525:{\displaystyle (a,b)}
2495:
2439:
2419:
2417:{\displaystyle (a,b)}
2363:
2247:
2143:
2043:
1954:
1892:
1809:
1714:
1594:
1297:), which, therefore,
1166:Joseph-Louis Lagrange
1137:
1053:is a given function.
1015:
797:invention of calculus
734:differential equation
647:Augustin-Louis Cauchy
632:Joseph-Louis Lagrange
464:Numerical integration
446:Exponential stability
309:Relation to processes
5579:Table of derivatives
5326:Finite volume method
5250:Dirac delta function
5219:Asymptotic stability
5161:Existence/uniqueness
5026:Integro-differential
4860:Informal mathematics
4740:Mathematical physics
4735:Mathematical finance
4720:Mathematical biology
4659:Diophantine geometry
4278:, S.O.S. Mathematics
4254:at Wikimedia Commons
4224:. Elsevier Science.
4143:Johnson, W. (1913).
4134:Ince, E. L. (1956).
4112:; Meade, D. (2017).
3991:Schelter, William F.
3887:, accessed Dec 2015.
3681:, Berlin, New York:
3628:. Cengage Learning.
3516:ode2(equation, y, x)
3388:difference equations
3335:quantum field theory
3261:
3234:
3214:
3162:
3126:
2956:
2792:
2778:for other results.)
2758:
2738:
2701:
2666:
2640:
2614:
2556:
2536:
2504:
2448:
2428:
2396:
2266:
2162:
2083:
1977:
1907:
1828:
1729:
1661:
1559:
1311:independent variable
1309:is often called the
1190:Lagrangian mechanics
1077:
1066:in 1695. This is an
1023:In all these cases,
820:
469:Dirac delta function
205:Integro-differential
5659:Continuous function
5612:Functional analysis
5336:Perturbation theory
5316:Runge–Kutta methods
5296:Integral transforms
5229:Rate of convergence
5125:(discrete analogue)
4875:Mathematics and art
4785:Operations research
4540:Functional analysis
4090:. pp. 266–277.
3906:Extract of page 247
3750:1987AmJPh..55...33W
3592:Recurrence relation
3404:applied mathematics
3382:Time scale calculus
3096:
3047:
2655:{\displaystyle x=a}
2629:{\displaystyle y=b}
2383:algebraic equations
1820:harmonic oscillator
1537:polynomial equation
1511:order of derivative
1494:harmonic oscillator
1393:partial derivatives
1225:classical mechanics
565:Perturbation theory
560:Integral transforms
451:Rate of convergence
317:(discrete analogue)
154:Population dynamics
121:Continuum mechanics
112:Applied mathematics
32:Difference equation
5691:Mathematics portal
5574:Lists of integrals
5457:Sofya Kovalevskaya
5291:Integrating factor
5214:Lyapunov stability
5134:Stochastic partial
4820:Numerical analysis
4429:Mathematical logic
4424:Information theory
4307:2008-12-19 at the
3883:2016-04-01 at the
3777:2020-02-09 at the
3571:Integral equations
3320:stochastic process
3298:Integral equations
3267:
3247:
3220:
3200:
3148:
3109:
3107:
3084:
3035:
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2744:
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2687:
2652:
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2600:
2542:
2522:
2490:
2434:
2414:
2358:
2242:
2138:
2038:
1949:
1887:
1804:
1709:
1604:thin-film equation
1589:
1352:Holonomic function
1237:equation of motion
1150:musical instrument
1132:
1010:
1008:
555:Integrating factor
396:Initial conditions
331:Stochastic partial
5698:
5697:
5664:Special functions
5627:Harmonic analysis
5465:
5464:
5344:
5343:
5149:
5148:
4946:
4945:
4545:Harmonic analysis
4250:Media related to
4231:978-1-4832-6396-0
4210:978-0-8218-8328-0
3947:www.maplesoft.com
3859:Weisstein, Eric W
3692:978-3-540-56670-0
3635:978-1-285-40110-2
3566:Initial condition
3543:desolve(y'=k*y,y)
3420:numerical methods
3354:ultrametric space
3270:{\displaystyle y}
3223:{\displaystyle g}
2894:
2846:
2767:{\displaystyle a}
2747:{\displaystyle Z}
2722:
2577:
2545:{\displaystyle Z}
2437:{\displaystyle Z}
2353:
2316:
2287:
2234:
2197:
2130:
2104:
2015:
1928:
1863:
1790:
1764:
1682:
1545:polynomial degree
1436:dynamical systems
1432:quantum mechanics
1363:numerical methods
1348:special functions
998:
954:
888:
845:
805:Gottfried Leibniz
785:numerical methods
779:puts emphasis on
726:
725:
617:Gottfried Leibniz
508:Finite difference
300:
299:
161:
160:
131:Dynamical systems
16:(Redirected from
5718:
5617:Fourier analysis
5597:Complex analysis
5498:Major topics in
5492:
5485:
5478:
5469:
5442:Phyllis Nicolson
5422:Rudolf Lipschitz
5259:Solution methods
5234:Series solutions
5158:
4991:
4973:
4966:
4959:
4950:
4934:
4933:
4922:
4921:
4910:
4909:
4899:
4898:
4830:Computer algebra
4805:Computer science
4525:Complex analysis
4359:
4352:
4345:
4336:
4271:Lamar University
4249:
4235:
4214:
4182:
4173:
4150:
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4117:
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4091:
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4057:
4056:
4041:
4035:
4034:
4032:
4031:
4025:doc.sagemath.org
4017:
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3869:
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3841:
3835:
3834:
3816:
3810:
3803:Speiser, David.
3801:
3795:
3788:
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3768:
3762:
3761:
3732:
3726:
3725:
3715:
3702:
3696:
3695:
3674:
3668:
3667:
3659:Bernoulli, Jacob
3655:
3649:
3646:
3640:
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3619:
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3526:
3517:
3508:
3499:
3281:Related concepts
3276:
3274:
3273:
3268:
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3254:
3253:
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3206:
3201:
3190:
3189:
3177:
3176:
3157:
3155:
3154:
3149:
3138:
3137:
3122:For any nonzero
3118:
3116:
3115:
3110:
3108:
3092:
3073:
3072:
3060:
3043:
3024:
3023:
3011:
2998:
2997:
2978:
2977:
2945:
2943:
2942:
2937:
2908:
2907:
2895:
2893:
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2877:
2866:
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2773:
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2765:
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2725:
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2705:
2696:
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2688:
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2214:
2213:
2203:
2198:
2196:
2195:
2194:
2181:
2177:
2176:
2166:
2154:Laplace equation
2147:
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2087:
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2012:
1999:
1995:
1994:
1984:
1958:
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1941:
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1911:
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1569:
1308:
1304:
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1288:
1284:
1162:Daniel Bernoulli
1141:
1139:
1138:
1133:
1130:
1129:
1087:
1052:
1048:
1039:
1030:
1026:
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928:
927:
889:
887:
879:
871:
846:
844:
836:
828:
718:
711:
704:
682:Phyllis Nicolson
667:Rudolf Lipschitz
550:Green's function
526:Infinite element
517:
482:Solution methods
460:
318:
229:By variable type
183:
65:Natural sciences
58:
37:
21:
5726:
5725:
5721:
5720:
5719:
5717:
5716:
5715:
5701:
5700:
5699:
5694:
5683:
5632:P-adic analysis
5583:
5569:Matrix calculus
5564:Tensor calculus
5559:Vector calculus
5522:Differentiation
5502:
5496:
5466:
5461:
5402:Jacob Bernoulli
5375:
5340:
5331:Galerkin method
5254:
5192:Solution topics
5187:
5145:
5111:
5050:
4982:
4977:
4947:
4942:
4893:
4884:
4834:
4791:
4770:Systems science
4701:
4697:Homotopy theory
4663:
4630:
4582:
4554:
4501:
4448:
4419:Category theory
4405:
4370:
4363:
4309:Wayback Machine
4242:
4232:
4217:
4211:
4185:
4176:
4170:
4157:
4142:
4133:
4120:
4107:
4095:Blanchard, P.;
4094:
4085:
4082:
4080:Further reading
4077:
4069:
4065:
4064:
4060:
4043:
4042:
4038:
4029:
4027:
4019:
4018:
4014:
3989:
3988:
3984:
3975:
3973:
3971:www.wolfram.com
3965:
3964:
3960:
3951:
3949:
3941:
3940:
3936:
3929:
3916:
3915:
3911:
3896:
3895:
3891:
3885:Wayback Machine
3876:
3872:
3857:
3853:
3843:
3842:
3838:
3818:
3817:
3813:
3802:
3798:
3789:
3785:
3779:Wayback Machine
3769:
3765:
3758:10.1119/1.15311
3734:
3733:
3729:
3713:
3704:
3703:
3699:
3693:
3683:Springer-Verlag
3676:
3675:
3671:
3664:Acta Eruditorum
3657:
3656:
3652:
3647:
3643:
3636:
3621:
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3607:
3551:
3542:
3533:
3524:
3515:
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3497:
3483:
3396:
3384:
3378:
3344:An ultrametric
3283:
3259:
3258:
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3160:
3159:
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3077:
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3051:
3028:
3015:
3004:
3002:
2989:
2982:
2969:
2954:
2953:
2899:
2886:
2878:
2857:
2835:
2831:
2817:
2816:
2795:
2790:
2789:
2756:
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2186:
2182:
2168:
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2159:
2122:
2114:
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2080:
2004:
2000:
1986:
1985:
1975:
1974:
1933:
1920:
1912:
1905:
1904:
1868:
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1848:
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1727:
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1696:
1674:
1666:
1659:
1658:
1612:
1573:
1562:
1557:
1556:
1508:is the highest
1502:
1470:linear equation
1462:
1456:
1420:electrodynamics
1377:
1371:
1306:
1302:
1294:
1293:(often denoted
1286:
1282:
1268:
1260:Main articles:
1258:
1249:
1221:
1152:was studied by
1121:
1080:
1075:
1074:
1060:Jacob Bernoulli
1050:
1047:
1041:
1038:
1032:
1028:
1024:
1007:
1006:
987:
983:
975:
963:
956:
943:
939:
931:
919:
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880:
872:
867:
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847:
837:
829:
818:
817:
793:
722:
693:
692:
691:
622:Jacob Bernoulli
606:
593:
592:
574:
543:Petrov–Galerkin
511:
496:
483:
475:
474:
473:
455:
401:Boundary values
390:
382:
381:
357:
344:
343:
342:
316:
310:
302:
301:
289:
266:
224:
180:
167:
166:
162:
140:Social sciences
96:
74:
55:
35:
28:
23:
22:
15:
12:
11:
5:
5724:
5722:
5714:
5713:
5703:
5702:
5696:
5695:
5688:
5685:
5684:
5682:
5681:
5676:
5671:
5666:
5661:
5656:
5650:
5649:
5644:
5642:Measure theory
5639:
5636:P-adic numbers
5629:
5624:
5619:
5614:
5609:
5599:
5594:
5588:
5585:
5584:
5582:
5581:
5576:
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5460:
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5444:
5439:
5434:
5429:
5424:
5419:
5417:Ernst Lindelöf
5414:
5409:
5404:
5399:
5397:Leonhard Euler
5394:
5389:
5383:
5381:
5380:Mathematicians
5377:
5376:
5374:
5373:
5368:
5363:
5358:
5352:
5350:
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5288:
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5204:Phase portrait
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5119:
5117:
5113:
5112:
5110:
5109:
5107:On jet bundles
5104:
5099:
5094:
5089:
5084:
5079:
5074:
5072:Nonhomogeneous
5069:
5064:
5058:
5056:
5052:
5051:
5049:
5048:
5043:
5038:
5033:
5028:
5023:
5018:
5013:
5008:
5003:
4997:
4995:
4988:
4987:Classification
4984:
4983:
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4976:
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4848:Mathematicians
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4840:Related topics
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4775:Control theory
4767:
4762:
4757:
4752:
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4742:
4737:
4732:
4727:
4722:
4717:
4711:
4709:
4703:
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4550:Measure theory
4547:
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4522:
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4364:
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4333:
4332:
4327:
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4299:
4294:
4285:
4279:
4273:
4269:Paul Dawkins,
4264:
4255:
4241:
4240:External links
4238:
4237:
4236:
4230:
4215:
4209:
4187:Teschl, Gerald
4183:
4174:
4168:
4155:
4140:
4131:
4129:. McGraw-Hill.
4118:
4105:
4097:Devaney, R. L.
4092:
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4076:
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4058:
4051:. 2022-08-22.
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3720:. New Series.
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3482:
3479:
3456:Joseph Fourier
3395:
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3374:
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3372:
3365:
3350:p-adic numbers
3342:
3327:
3324:Wiener process
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1498:
1490:approximations
1458:Main article:
1455:
1452:
1416:electrostatics
1401:computer model
1373:Main article:
1370:
1367:
1257:
1254:
1248:
1245:
1220:
1217:
1158:Leonhard Euler
1143:
1142:
1128:
1124:
1120:
1117:
1114:
1111:
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792:
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689:
684:
679:
674:
672:Ernst Lindelöf
669:
664:
659:
654:
649:
644:
642:Joseph Fourier
639:
634:
629:
627:Leonhard Euler
624:
619:
614:
608:
607:
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603:
600:
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595:
594:
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580:
573:
572:
567:
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535:
530:
529:
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521:Finite element
518:
514:Crank–Nicolson
505:
500:
494:
489:
485:
484:
481:
480:
477:
476:
472:
471:
466:
461:
453:
448:
435:
433:Phase portrait
430:
425:
424:
423:
421:Cauchy problem
418:
413:
408:
398:
392:
391:
389:General topics
388:
387:
384:
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380:
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369:
364:
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350:
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345:
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298:
297:
296:
295:
288:
287:
282:
276:
273:
272:
268:
267:
265:
264:
262:Nonhomogeneous
255:
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241:
240:
239:
231:
230:
226:
225:
223:
222:
217:
212:
207:
202:
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192:
186:
181:
178:
177:
174:
173:
172:Classification
169:
168:
159:
158:
157:
156:
151:
143:
142:
136:
135:
134:
133:
128:
123:
115:
114:
108:
107:
106:
105:
100:
94:
89:
84:
76:
75:
73:
72:
67:
61:
56:
53:
52:
49:
48:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5723:
5712:
5709:
5708:
5706:
5693:
5692:
5686:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5651:
5648:
5645:
5643:
5640:
5637:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5607:
5603:
5600:
5598:
5595:
5593:
5592:Real analysis
5590:
5589:
5586:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5555:
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5540:
5538:
5535:
5533:
5530:
5529:
5528:
5525:
5523:
5520:
5518:
5514:
5513:
5509:
5508:
5505:
5501:
5493:
5488:
5486:
5481:
5479:
5474:
5473:
5470:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5384:
5382:
5378:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5354:
5353:
5351:
5347:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5263:
5261:
5257:
5251:
5248:
5246:
5243:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5210:
5207:
5205:
5202:
5200:
5197:
5196:
5194:
5190:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5165:
5163:
5159:
5156:
5152:
5142:
5139:
5135:
5132:
5131:
5130:
5127:
5124:
5121:
5120:
5118:
5114:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5059:
5057:
5053:
5047:
5044:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4992:
4989:
4985:
4981:
4974:
4969:
4967:
4962:
4960:
4955:
4954:
4951:
4939:
4938:
4929:
4927:
4926:
4917:
4915:
4914:
4905:
4903:
4902:
4897:
4891:
4890:
4887:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4854:
4851:
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4845:
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4841:
4837:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4806:
4803:
4802:
4800:
4798:
4797:Computational
4794:
4786:
4783:
4781:
4778:
4776:
4773:
4772:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4712:
4710:
4708:
4704:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4674:
4672:
4670:
4666:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4641:
4639:
4637:
4636:Number theory
4633:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4593:
4591:
4589:
4585:
4579:
4576:
4574:
4571:
4569:
4568:Combinatorics
4566:
4565:
4563:
4561:
4557:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4520:Real analysis
4518:
4516:
4513:
4512:
4510:
4508:
4504:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4459:
4457:
4455:
4451:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4416:
4414:
4412:
4408:
4402:
4399:
4397:
4394:
4390:
4387:
4385:
4382:
4381:
4380:
4377:
4376:
4373:
4368:
4360:
4355:
4353:
4348:
4346:
4341:
4340:
4337:
4331:
4328:
4325:
4322:
4320:
4316:
4313:
4311:MATLAB models
4310:
4306:
4303:
4300:
4298:
4295:
4293:
4289:
4286:
4283:
4280:
4277:
4274:
4272:
4268:
4265:
4262:
4259:
4256:
4253:
4248:
4244:
4243:
4239:
4233:
4227:
4223:
4222:
4216:
4212:
4206:
4202:
4198:
4194:
4193:
4188:
4184:
4180:
4175:
4171:
4169:1-58488-297-2
4165:
4161:
4156:
4154:
4148:
4147:
4141:
4137:
4132:
4127:
4126:
4119:
4115:
4111:
4106:
4102:
4098:
4093:
4089:
4084:
4083:
4079:
4068:
4062:
4059:
4054:
4050:
4046:
4040:
4037:
4026:
4022:
4016:
4013:
4008:
4004:
4002:
3996:
3992:
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3983:
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3938:
3935:
3930:
3928:0-534-37388-7
3924:
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3824:
3823:
3815:
3812:
3808:
3807:
3800:
3797:
3793:
3787:
3784:
3780:
3776:
3773:
3767:
3764:
3759:
3755:
3751:
3747:
3743:
3740:
3739:
3731:
3728:
3723:
3719:
3712:
3710:
3701:
3698:
3694:
3688:
3684:
3680:
3673:
3670:
3666:
3665:
3660:
3654:
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3645:
3642:
3637:
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3627:
3626:
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3609:
3604:
3601:
3599:
3596:
3593:
3590:
3587:
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3582:
3579:
3577:
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3569:
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3504:
3501:
3495:
3492:
3491:
3490:
3488:
3480:
3478:
3476:
3471:
3469:
3468:Black–Scholes
3465:
3461:
3460:heat equation
3457:
3453:
3452:wave equation
3449:
3444:
3440:
3436:
3432:
3428:
3423:
3421:
3417:
3413:
3409:
3405:
3401:
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3391:
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3366:
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3336:
3332:
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3325:
3321:
3317:
3313:
3310:
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3303:
3299:
3296:
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3285:
3284:
3280:
3278:
3264:
3242:
3238:
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3186:
3182:
3178:
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3097:
3093:
3089:
3085:
3081:
3079:
3069:
3065:
3057:
3054:
3048:
3044:
3040:
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3030:
3020:
3016:
3008:
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2994:
2990:
2986:
2984:
2974:
2970:
2963:
2952:
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2930:
2924:
2921:
2918:
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2896:
2890:
2887:
2882:
2879:
2870:
2862:
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2854:
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2836:
2832:
2827:
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2808:
2800:
2796:
2788:
2787:
2786:
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2779:
2777:
2761:
2741:
2718:
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2675:
2669:
2649:
2646:
2643:
2623:
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2591:
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2570:
2565:
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2513:
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2478:
2472:
2466:
2463:
2460:
2454:
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2431:
2408:
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2391:
2386:
2384:
2380:
2373:
2355:
2347:
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2334:
2329:
2318:
2312:
2304:
2295:
2292:
2289:
2283:
2275:
2262:
2261:
2259:
2255:
2239:
2236:
2228:
2224:
2215:
2210:
2199:
2191:
2187:
2178:
2173:
2158:
2157:
2155:
2151:
2135:
2132:
2126:
2118:
2109:
2106:
2100:
2092:
2079:
2078:
2076:
2075:
2074:
2072:
2068:
2064:
2060:
2056:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2009:
2005:
2001:
1996:
1991:
1987:
1980:
1973:
1972:
1970:
1966:
1962:
1946:
1943:
1938:
1934:
1930:
1924:
1921:
1916:
1913:
1903:
1902:
1900:
1884:
1881:
1878:
1873:
1869:
1865:
1857:
1853:
1849:
1844:
1839:
1835:
1824:
1823:
1821:
1817:
1801:
1798:
1795:
1792:
1786:
1783:
1778:
1775:
1769:
1766:
1758:
1754:
1750:
1745:
1740:
1736:
1725:
1724:
1722:
1706:
1701:
1697:
1693:
1690:
1687:
1684:
1678:
1675:
1670:
1667:
1657:
1656:
1654:
1653:
1652:
1650:
1649:heterogeneous
1646:
1644:
1639:
1635:
1634:
1629:
1625:
1621:
1617:
1609:
1607:
1605:
1600:
1586:
1583:
1578:
1574:
1570:
1566:
1563:
1554:
1550:
1546:
1542:
1538:
1533:
1532:, and so on.
1531:
1527:
1523:
1522:
1517:
1513:
1512:
1507:
1499:
1497:
1495:
1491:
1486:
1484:
1480:
1476:
1471:
1467:
1461:
1453:
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1449:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1404:
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1394:
1390:
1386:
1382:
1376:
1368:
1366:
1364:
1360:
1355:
1353:
1349:
1345:
1340:
1338:
1334:
1330:
1326:
1324:
1320:
1316:
1312:
1300:
1292:
1281:
1277:
1273:
1267:
1263:
1255:
1253:
1246:
1244:
1240:
1238:
1233:
1230:
1229:Newton's laws
1226:
1218:
1216:
1214:
1213:heat equation
1210:
1206:
1202:
1198:
1193:
1191:
1187:
1182:
1178:
1173:
1171:
1170:wave equation
1167:
1163:
1159:
1155:
1151:
1146:
1126:
1122:
1115:
1109:
1106:
1103:
1097:
1091:
1088:
1084:
1081:
1073:
1072:
1071:
1069:
1065:
1062:proposed the
1061:
1057:
1054:
1044:
1035:
1003:
1000:
992:
988:
979:
968:
964:
960:
958:
948:
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935:
924:
920:
909:
906:
903:
897:
894:
892:
884:
881:
876:
873:
860:
854:
851:
849:
841:
838:
833:
830:
816:
815:
814:
812:
811:
806:
802:
798:
790:
788:
786:
782:
778:
774:
771:Often when a
769:
765:
763:
759:
755:
751:
747:
743:
739:
735:
731:
719:
714:
712:
707:
705:
700:
699:
697:
696:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
596:
589:
586:
584:
581:
579:
576:
575:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
544:
541:
540:
539:
536:
534:
533:Finite volume
531:
527:
524:
523:
522:
519:
515:
509:
506:
504:
501:
499:
495:
493:
490:
487:
486:
479:
478:
470:
467:
465:
462:
458:
454:
452:
449:
447:
443:
439:
436:
434:
431:
429:
426:
422:
419:
417:
414:
412:
409:
407:
404:
403:
402:
399:
397:
394:
393:
386:
385:
378:
375:
373:
370:
368:
365:
363:
360:
359:
353:
352:
347:
339:
336:
332:
329:
328:
327:
324:
323:
319:
313:
312:
306:
305:
294:
291:
290:
286:
283:
281:
278:
277:
275:
274:
269:
263:
259:
256:
254:
251:
248:
246:
243:
242:
238:
235:
234:
233:
232:
227:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
187:
185:
184:
176:
175:
170:
165:
155:
152:
150:
147:
146:
145:
144:
141:
137:
132:
129:
127:
124:
122:
119:
118:
117:
116:
113:
109:
104:
101:
99:
95:
93:
90:
88:
85:
83:
80:
79:
78:
77:
71:
68:
66:
63:
62:
59:
51:
50:
45:
42:
38:
33:
19:
5689:
5526:
5510:
5452:Martin Kutta
5407:Émile Picard
5387:Isaac Newton
5301:Euler method
5271:Substitution
5091:
5086:
5081:
5076:
4979:
4935:
4923:
4911:
4892:
4825:Optimization
4687:Differential
4611:Differential
4578:Order theory
4573:Graph theory
4534:
4477:Group theory
4220:
4191:
4178:
4159:
4145:
4135:
4124:
4113:
4100:
4087:
4061:
4048:
4039:
4028:. Retrieved
4024:
4015:
3998:
3985:
3974:. Retrieved
3970:
3961:
3950:. Retrieved
3946:
3937:
3918:
3912:
3899:
3892:
3873:
3854:
3845:
3839:
3821:
3814:
3804:
3799:
3786:
3766:
3744:(1): 33–37.
3741:
3738:Am. J. Phys.
3736:
3730:
3721:
3717:
3708:
3700:
3678:
3672:
3662:
3653:
3644:
3624:
3617:
3484:
3472:
3424:
3397:
3394:Applications
3385:
3291:
3121:
2948:
2780:
2444:, such that
2387:
2377:
2258:KdV equation
2070:
2066:
2062:
2058:
2054:
2052:
1968:
1648:
1642:
1637:
1631:
1627:
1623:
1619:
1615:
1613:
1601:
1549:total degree
1540:
1534:
1529:
1519:
1509:
1505:
1503:
1487:
1465:
1463:
1405:
1384:
1378:
1356:
1341:
1327:
1322:
1314:
1298:
1275:
1269:
1250:
1241:
1234:
1222:
1204:
1194:
1174:
1147:
1144:
1070:of the form
1058:
1055:
1042:
1033:
1022:
809:
801:Isaac Newton
794:
770:
766:
733:
727:
677:Émile Picard
662:Martin Kutta
652:George Green
612:Isaac Newton
444: /
440: /
260: /
126:Chaos theory
40:
5517:Integration
5209:Phase space
5067:Homogeneous
4937:WikiProject
4780:Game theory
4760:Probability
4497:Homological
4487:Multilinear
4467:Commutative
4444:Type theory
4411:Foundations
4367:mathematics
4110:DiPrima, R.
4108:Boyce, W.;
4103:. Thompson.
4001:SourceForge
3707:"Review of
3503:Mathematica
3416:closed form
3412:engineering
3360:instead of
1643:homogeneous
1181:tautochrone
781:qualitative
750:engineering
746:derivatives
730:mathematics
570:Runge–Kutta
315:Difference
258:Homogeneous
70:Engineering
5542:stochastic
5437:John Crank
5266:Inspection
5129:Stochastic
5123:Difference
5097:Autonomous
5041:Non-linear
5031:Fractional
4994:Operations
4765:Statistics
4644:Arithmetic
4606:Arithmetic
4472:Elementary
4439:Set theory
4197:Providence
4030:2020-05-09
3976:2020-06-28
3952:2020-05-09
3610:References
3380:See also:
2949:such that
1967:of length
1475:symmetries
1448:randomness
1428:elasticity
1424:fluid flow
1391:and their
744:and their
687:John Crank
488:Inspection
442:Asymptotic
326:Stochastic
245:Autonomous
220:Non-linear
210:Fractional
5654:Functions
5241:solutions
5199:Wronskian
5154:Solutions
5082:Decoupled
5046:Holonomic
4692:Geometric
4682:Algebraic
4621:Euclidean
4596:Algebraic
4492:Universal
3863:MathWorld
3525:desolve()
3464:diffusion
3439:economics
3431:chemistry
3302:integrals
3195:…
3103:…
2852:⋯
2716:∂
2708:∂
2473:×
2340:∂
2326:∂
2319:−
2310:∂
2302:∂
2281:∂
2273:∂
2221:∂
2207:∂
2184:∂
2170:∂
2124:∂
2116:∂
2098:∂
2090:∂
2027:
1870:ω
1767:−
1638:nonlinear
1337:integrals
1323:more than
1201:heat flow
1195:In 1822,
1186:mechanics
985:∂
977:∂
941:∂
933:∂
758:economics
742:functions
428:Wronskian
406:Dirichlet
149:Economics
92:Chemistry
82:Astronomy
5705:Category
5679:Infinity
5532:ordinary
5512:Calculus
5349:Examples
5239:Integral
5011:Ordinary
4913:Category
4669:Topology
4616:Discrete
4601:Analytic
4588:Geometry
4560:Discrete
4515:Calculus
4507:Analysis
4462:Abstract
4401:Glossary
4384:Timeline
4305:Archived
4189:(2012).
4138:. Dover.
4116:. Wiley.
4053:Archived
4007:Archived
3881:Archived
3775:Archived
3549:See also
3521:SageMath
3481:Software
3094:″
3058:″
3045:′
3009:′
1965:pendulum
1610:Examples
1567:′
1315:ordinary
1291:variable
1085:′
738:equation
538:Galerkin
438:Lyapunov
349:Solution
293:Notation
285:Operator
271:Features
190:Ordinary
5537:partial
5077:Coupled
5016:Partial
4925:Commons
4707:Applied
4677:General
4454:Algebra
4379:History
3831:2688081
3746:Bibcode
3435:biology
3427:physics
3408:physics
2379:Solving
1344:physics
1305:. Thus
1299:depends
1219:Example
1197:Fourier
1049:), and
1031:(or of
791:History
762:biology
754:physics
411:Neumann
195:Partial
103:Geology
98:Biology
87:Physics
5674:Series
5092:Degree
5036:Linear
4626:Finite
4482:Linear
4389:Future
4365:Major
4292:Maxima
4228:
4207:
4166:
3925:
3829:
3689:
3632:
3512:Maxima
3507:DSolve
3498:dsolve
3450:, the
3410:, and
3352:in an
1651:ones.
1633:linear
1622:, and
1333:linear
1164:, and
760:, and
736:is an
598:People
510:
457:Series
215:Linear
54:Fields
5669:Limit
5141:Delay
5087:Order
4853:lists
4396:Lists
4369:areas
4070:(PDF)
4045:"ODE"
3714:(PDF)
3530:SymPy
3494:Maple
3485:Some
3443:model
3158:, if
2636:when
1528:is a
1518:is a
1479:chaos
1430:, or
1408:sound
1247:Types
498:Euler
416:Robin
338:Delay
280:Order
253:Exact
179:Types
47:Scope
4319:UIUC
4226:ISBN
4205:ISBN
4164:ISBN
3923:ISBN
3827:OCLC
3724:(1).
3687:ISBN
3630:ISBN
3539:Xcas
3437:and
3429:and
3402:and
3400:pure
3337:and
3292:time
3210:and
2697:and
2500:and
2069:and
2061:and
1647:and
1636:and
1626:and
1504:The
1412:heat
1264:and
1175:The
1040:and
803:and
732:, a
605:List
4261:MIT
4151:In
3754:doi
3487:CAS
3307:An
2065:or
2024:sin
1385:PDE
1354:).
1301:on
1276:ODE
1270:An
1223:In
1203:in
799:by
728:In
5707::
5515::
4203:.
4199::
4195:.
4047:.
4023:.
4005:.
3997:.
3969:.
3945:.
3752:.
3742:55
3716:.
3685:,
3541::
3532::
3523::
3514::
3505::
3496::
3477:.
3422:.
3406:,
3367:A
3329:A
3314:A
3286:A
3257:,
2260::
2240:0.
2156::
2136:0.
2073:.
2036:0.
1971::
1947:4.
1885:0.
1822::
1802:0.
1464:A
1450:.
1442:.
1426:,
1422:,
1418:,
1414:,
1410:,
1403:.
1379:A
1361:,
1339:.
1192:.
1160:,
1156:,
764:.
756:,
752:,
5638:)
5634:(
5608:)
5604:(
5491:e
5484:t
5477:v
4972:e
4965:t
4958:v
4358:e
4351:t
4344:v
4234:.
4213:.
4181:.
4172:.
4072:.
4033:.
4003:)
3979:.
3955:.
3931:.
3833:.
3760:.
3756::
3748::
3722:9
3638:.
3364:.
3341:.
3304:.
3265:y
3243:0
3239:x
3218:g
3198:}
3192:,
3187:1
3183:f
3179:,
3174:0
3170:f
3166:{
3146:)
3143:x
3140:(
3135:n
3131:f
3098:,
3090:0
3086:y
3082:=
3075:)
3070:0
3066:x
3062:(
3055:y
3049:,
3041:0
3037:y
3033:=
3026:)
3021:0
3017:x
3013:(
3006:y
3000:,
2995:0
2991:y
2987:=
2980:)
2975:0
2971:x
2967:(
2964:y
2934:)
2931:x
2928:(
2925:g
2922:=
2919:y
2916:)
2913:x
2910:(
2905:0
2901:f
2897:+
2891:x
2888:d
2883:y
2880:d
2874:)
2871:x
2868:(
2863:1
2859:f
2855:+
2849:+
2841:n
2837:x
2833:d
2828:y
2823:n
2819:d
2812:)
2809:x
2806:(
2801:n
2797:f
2762:a
2742:Z
2719:x
2711:g
2685:)
2682:y
2679:,
2676:x
2673:(
2670:g
2650:a
2647:=
2644:x
2624:b
2621:=
2618:y
2598:)
2595:y
2592:,
2589:x
2586:(
2583:g
2580:=
2574:x
2571:d
2566:y
2563:d
2540:Z
2520:)
2517:b
2514:,
2511:a
2508:(
2488:]
2485:p
2482:,
2479:n
2476:[
2470:]
2467:m
2464:,
2461:l
2458:[
2455:=
2452:Z
2432:Z
2412:)
2409:b
2406:,
2403:a
2400:(
2356:.
2348:3
2344:x
2335:u
2330:3
2313:x
2305:u
2296:u
2293:6
2290:=
2284:t
2276:u
2237:=
2229:2
2225:y
2216:u
2211:2
2200:+
2192:2
2188:x
2179:u
2174:2
2133:=
2127:x
2119:u
2110:t
2107:+
2101:t
2093:u
2071:y
2067:x
2063:t
2059:x
2055:u
2033:=
2030:u
2021:g
2018:+
2010:2
2006:x
2002:d
1997:u
1992:2
1988:d
1981:L
1969:L
1944:+
1939:2
1935:u
1931:=
1925:x
1922:d
1917:u
1914:d
1882:=
1879:u
1874:2
1866:+
1858:2
1854:x
1850:d
1845:u
1840:2
1836:d
1799:=
1796:u
1793:+
1787:x
1784:d
1779:u
1776:d
1770:x
1759:2
1755:x
1751:d
1746:u
1741:2
1737:d
1707:.
1702:2
1698:x
1694:+
1691:u
1688:c
1685:=
1679:x
1676:d
1671:u
1668:d
1628:ω
1624:c
1620:x
1616:u
1587:0
1584:=
1579:2
1575:y
1571:+
1564:y
1383:(
1307:x
1303:x
1295:y
1287:x
1283:x
1274:(
1127:n
1123:y
1119:)
1116:x
1113:(
1110:Q
1107:=
1104:y
1101:)
1098:x
1095:(
1092:P
1089:+
1082:y
1051:f
1046:2
1043:x
1037:1
1034:x
1029:x
1025:y
1004:y
1001:=
993:2
989:x
980:y
969:2
965:x
961:+
949:1
945:x
936:y
925:1
921:x
913:)
910:y
907:,
904:x
901:(
898:f
895:=
885:x
882:d
877:y
874:d
864:)
861:x
858:(
855:f
852:=
842:x
839:d
834:y
831:d
717:e
710:t
703:v
516:)
512:(
34:.
20:)
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