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{{Short description|Type of functional equation (mathematics)}}
{{Distinguish|Difference equation}}
{{Differential equations}}

In [[mathematics]], a '''differential equation''' is an [[functional equation|equation]] that relates one or more unknown [[function (mathematics)|function]]s and their [[derivative]]s.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=%22ordinary+differential%22|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2}}</ref> 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 in [[Mathematical model|mathematical models]] and [[Scientific law|scientific laws]]; therefore, differential equations play a prominent role in many disciplines including [[engineering]], [[physics]], [[economics]], and [[biology]].

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 a [[closed-form expression]] for the solutions is not available, solutions may be approximated numerically using computers, and many [[numerical methods]] have been developed to determine solutions with a given degree of accuracy. The [[Dynamical systems theory|theory of dynamical systems]] analyzes the [[:wikt:qualitative|qualitative]] aspects of solutions, such as their [[Ergodic theory|average behavior]] over a long time interval.

==History==

Differential equations came into existence with the [[History of calculus|invention of calculus]] by [[Isaac Newton]] and [[Gottfried Leibniz]]. In Chapter 2 of his 1671 work [[Method of Fluxions|''Methodus fluxionum et Serierum Infinitarum'']],<ref>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].</ref> Newton listed three kinds of differential equations:

:<math>\begin{align}
  \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{align}</math>
In all these cases, {{mvar|y}} is an unknown function of {{mvar|x}} (or of {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}}), and {{mvar|f}} is a given function.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

[[Jacob Bernoulli]] proposed the [[Bernoulli differential equation]] in 1695.<ref>{{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=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 | year=1695 | journal=[[Acta Eruditorum]]}}</ref> This is an [[ordinary differential equation]] of the form

: <math>y'+ P(x)y = Q(x)y^n\,</math>

for which the following year Leibniz obtained solutions by simplifying it.<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>

Historically, the problem of a vibrating string such as that of a [[musical instrument]] was studied by [[Jean le Rond d'Alembert]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph-Louis Lagrange]].<ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin of the American Mathematical Society |series=New Series |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first1=Gerard F. |last1=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] {{Webarchive|url=https://web.archive.org/web/20200209023122/http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= |date=2020-02-09 }} (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> In 1746, d’Alembert discovered the one-dimensional [[wave equation]], and within ten years Euler discovered the three-dimensional wave equation.<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref>

The [[Euler–Lagrange equation]] was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] 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 [[mechanics]], which led to the formulation of [[Lagrangian mechanics]].

In 1822, [[Joseph Fourier|Fourier]] published his work on [[heat flow]] in ''Théorie analytique de la chaleur'' (The Analytic Theory of Heat),<ref>{{Cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = fr | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref> in which he based his reasoning on [[Newton'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 his [[heat equation]] for conductive diffusion of heat. This partial differential equation is now a common part of mathematical physics curriculum.

==Example==
In [[classical mechanics]], the motion of a body is described by its position and velocity as the time value varies. [[Newton's laws of motion|Newton's laws]] 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.

In some cases, this differential equation (called an [[equations of motion|equation 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==
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 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===
{{main|Ordinary differential equation|Linear differential equation}}
An [[ordinary differential equation]] (''ODE'') is an equation containing an unknown [[function of a real variable|function of one real or complex variable]] {{mvar|x}}, its derivatives, and some given functions of {{mvar|x}}. The unknown function is generally represented by a [[variable (mathematics)|variable]] (often denoted {{mvar|y}}), which, therefore, ''depends'' on {{mvar|x}}. Thus {{mvar|x}} is often called the [[independent variable]] of the equation. The term "''ordinary''" is used in contrast with the term [[partial differential equation]], which may be with respect to ''more than'' one independent variable.

[[Linear differential equation]]s are the differential equations that are [[linear equation|linear]] in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of [[antiderivative|integrals]].

Most ODEs that are encountered in [[physics]] are linear. Therefore, most [[special functions]] may be defined as solutions of linear differential equations (see [[Holonomic function]]).

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

===Partial differential equations===
{{main|Partial differential equation}}
A [[partial differential equation]] (''PDE'') is a differential equation that contains unknown [[Multivariable calculus|multivariable function]]s and their [[partial derivatives]]. (This is in contrast to [[ordinary differential equations]], 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 [[computer model]].

PDEs can be used to describe a wide variety of phenomena in nature such as [[sound]], [[heat]], [[electrostatics]], [[electrodynamics]], [[fluid flow]], [[Elasticity (physics)|elasticity]], or [[quantum mechanics]]. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional [[dynamical systems]], partial differential equations often model [[multidimensional systems]]. [[Stochastic partial differential equations]] generalize partial differential equations for modeling [[randomness]].

===Non-linear differential equations===
{{main|Non-linear differential equations}}
A '''non-linear differential equation''' is a differential equation that is not a [[linear 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 particular [[Symmetry|symmetries]]. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of [[chaos theory|chaos]]. 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. [[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.<ref>{{cite book
 | last1 = Boyce
 | first1 = William E.
 | last2 = DiPrima
 | first2 = Richard C.
 | title = Elementary Differential Equations and Boundary Value Problems
 | publisher =John Wiley & Sons
 | edition = 4th
 | year = 1967
 | pages = 3
 }}</ref>

Linear differential equations frequently appear as [[linearization|approximations]] to nonlinear equations. These approximations are only valid under restricted conditions. For example, the [[harmonic oscillator]] equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations.

===Equation order and degree{{anchor|Second order|Order}}===
The '''order of the differential equation''' is the highest ''[[order of derivative]]'' of the unknown function that appears in the differential equation. 
For example, an equation containing only [[first-order derivative]]s is a ''[[first-order differential equation]]'', an equation containing the [[second-order derivative]] is a ''second-order differential equation'', and so on.<ref>[[Eric W Weisstein|Weisstein, Eric W]]. "Ordinary Differential Equation Order." From [[MathWorld]]--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html</ref><ref>{{usurped|1=[https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx Order and degree of a differential equation]}}, accessed Dec 2015.</ref>

When it is written as a [[polynomial equation]] in the unknown function and its derivatives, its '''degree of the differential equation''' is, depending on the context, the [[polynomial degree]] in the highest derivative of the unknown function,<ref>{{cite book |title=Elements of the Differential and Integral Calculus |author1=Elias Loomis |edition=revised |publisher=Harper & Bros. |year=1887 |isbn= |page=247 |url=https://books.google.com/books?id=DTI4AQAAMAAJ}} [https://books.google.com/books?id=DTI4AQAAMAAJ&pg=PA247 Extract of page 247]</ref> or its [[total degree]] in the unknown function and its derivatives. In particular, a [[linear differential equation]] has degree one for both meanings, but the non-linear differential equation <math>y'+y^2=0</math> is of degree one for the first meaning but not for the second one.

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 [[thin-film equation]], which is a fourth order partial differential equation.

===Examples===

In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between ''[[linear differential equation|linear]]'' and ''nonlinear'' differential equations, and between [[homogeneous differential equation|''homogeneous'' differential equation]]s and ''heterogeneous'' ones.
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
*: <math> \frac{du}{dx} = cu+x^2. </math>
* Homogeneous second-order linear ordinary differential equation:
*: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
*: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
* Heterogeneous first-order nonlinear ordinary differential equation:
*: <math> \frac{du}{dx} = u^2 + 4. </math>
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
*: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''.
* Homogeneous first-order linear partial differential equation:
*: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
*: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>
* Homogeneous third-order non-linear partial differential equation, the [[Korteweg–De Vries equation|KdV equation]]:
*: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>

==Existence of solutions==

[[Equation solving|Solving]] differential equations is not like solving [[algebraic equations]]. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, the [[Peano existence theorem]] gives one set of circumstances in which a solution exists. Given any point <math>(a, b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l, m]\times[n, p]</math> and <math>(a, b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math display="inline">\frac{dy}{dx} = g(x, y)</math> and the condition that <math>y = b</math> when <math>x = a</math>, then there is locally a solution to this problem if <math>g(x, y)</math> and <math display="inline">\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See [[Ordinary differential equation]] for other results.)

However, this only helps us with first order [[initial value problem]]s. Suppose we had a linear initial value problem of the nth order:
:<math>f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x)</math>

such that
:<math>\begin{align}
    y(x_{0}) &= y_{0}, &
   y'(x_{0}) &= y'_{0}, &
  y''(x_{0}) &= y''_{0}, &
      \ldots
\end{align}</math>

For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\ldots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> exists and is unique.<ref>{{cite book|last1=Zill|first1=Dennis G.|title=A First Course in Differential Equations|publisher=Brooks/Cole|isbn=0-534-37388-7|edition=5th|year=2001}}</ref>

==Related concepts==
* A [[delay differential equation]] (DDE) is an equation for a function of a single variable, usually called '''time''', in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
* [[Integral equation]]s may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains [[integral]]s.
* An [[integro-differential equation]] (IDE) is an equation that combines aspects of a differential equation and an integral equation.
* A [[stochastic differential equation]] (SDE) is an equation in which the unknown quantity is a [[stochastic process]] and the equation involves some known stochastic processes, for example, the [[Wiener process]] in the case of diffusion equations.
* A [[stochastic partial differential equation]] (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in [[quantum field theory]] and [[statistical mechanics]].
* An ultrametric [[pseudo-differential equation]] is an equation which contains [[p-adic numbers]] in an [[ultrametric space]]. Mathematical models that involve ultrametric pseudo-differential equations use [[pseudo-differential operators]] instead of [[differential operators]].
* A [[differential algebraic equation]] (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.

==Connection to difference equations==
{{See also|Time scale calculus}}

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 the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

==Applications==

The study of differential equations is a wide field in [[pure mathematics|pure]] and [[applied mathematics]], [[physics]], and [[engineering]]. 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 [[closed-form expression|closed form]] solutions. Instead, solutions can be approximated using [[Numerical ordinary differential equations|numerical methods]].

Many fundamental laws of [[physics]] and [[chemistry]] can be formulated as differential equations. In [[biology]] and [[economics]], differential equations are used to [[mathematical modelling|model]] the behavior of [[complex system]]s. 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. 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 [[partial differential equation]], the [[wave 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 by [[Joseph Fourier]], is governed by another second-order partial differential equation, the [[heat equation]]. It turns out that many [[diffusion]] processes, while seemingly different, are described by the same equation; the [[Black–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 is a witness of the importance of the topic. See [[List of named differential equations]].

== Software ==
Some [[Computer algebra system|CAS]] software can solve differential equations. These are the commands used in the leading programs:
* [[Maple (software)|Maple]]:<ref>{{Cite web|title=dsolve - Maple Programming Help|url=https://www.maplesoft.com/support/help/Maple/view.aspx?path=dsolve|website=www.maplesoft.com|access-date=2020-05-09}}</ref> <code>dsolve</code>
* [[Wolfram Mathematica|Mathematica]]:<ref>{{Cite web|title=DSolve - Wolfram Language Documentation|url=https://reference.wolfram.com/language/ref/DSolve.html|website=www.wolfram.com|access-date=2020-06-28}}</ref> <code>DSolve[]</code>
* [[Maxima (software)|Maxima]]:<ref>{{cite web |url=https://maxima.sourceforge.io/docs/tutorial/en/gaertner-tutorial-revision/Pages/ODE0001.htm |website=The Computer Algebra Program Maxima - a Tutorial (in Maxima documentation on [[SourceForge]]) |title=Differential Equations - Symbolic Solutions |first=William F. |last=Schelter |author-link=Bill Schelter |editor-first=Boris |editor-last=Gaertner |archive-url=https://web.archive.org/web/20221004221208/https://maxima.sourceforge.io/docs/tutorial/en/gaertner-tutorial-revision/Pages/ODE0001.htm |archive-date=2022-10-04 |url-status=live }}</ref> <code>ode2(equation, y, x)</code>
* [[SageMath]]:<ref>{{Cite web|title=Basic Algebra and Calculus — Sage Tutorial v9.0|url=http://doc.sagemath.org/html/en/tutorial/tour_algebra.html|website=doc.sagemath.org|access-date=2020-05-09}}</ref> <code>desolve()</code>
* [[SymPy]]:<ref>{{cite web |url=https://docs.sympy.org/latest/modules/solvers/ode.html |date=2022-08-22 |title=ODE |website=SymPy 1.11 documentation |archive-url=https://web.archive.org/web/20220926154502/https://docs.sympy.org/latest/modules/solvers/ode.html |archive-date=2022-09-26 |url-status=live }}</ref> <code>sympy.solvers.ode.dsolve(equation)</code>
* [[Xcas]]:<ref>{{Cite web|title=Symbolic algebra and Mathematics with Xcas|url=http://www-fourier.ujf-grenoble.fr/~parisse/giac/cascmd_en.pdf}}</ref> <code>desolve(y'=k*y,y)</code>

==See also==
{{Div col|colwidth=22em}}
* [[Exact differential equation]]
* [[Functional differential equation]]
* [[Initial condition]]
* [[Integral equations]]
* [[Numerical methods for ordinary differential equations]]
* [[Numerical methods for partial differential equations]]
* [[Picard–Lindelöf theorem]] on existence and uniqueness of solutions
* [[Recurrence relation]], also known as 'difference equation'
* [[Abstract differential equation]]
* [[System of differential equations]]
{{div col end}}

==References==
{{reflist|30em}}

==Further reading==
* {{cite book |first1=P. |last1=Abbott |first2=H. |last2=Neill |title=Teach Yourself Calculus |year=2003 |pages=266–277 }}
* {{cite book |first1=P. |last1=Blanchard |author-link2=Robert L. Devaney |first2=R. L. |last2=Devaney |first3=G. R. |last3=Hall |title=Differential Equations |publisher=Thompson |year=2006 }}
* {{cite book |first1=W. |last1=Boyce | first2=R. |last2=DiPrima|authorlink2=Richard DiPrima | first3=D. |last3=Meade| title=Elementary Differential Equations and Boundary Value Problems|publisher=Wiley |year=2017}}
* {{cite book |first1=E. A. |last1=Coddington |first2=N. |last2=Levinson |title=Theory of Ordinary Differential Equations |url=https://archive.org/details/theoryofordinary00codd |url-access=registration |publisher=McGraw-Hill |year=1955 }}
* {{cite book |first=E. L. |last=Ince |title=Ordinary Differential Equations |publisher=Dover |year=1956 }}
* {{cite book |first=W. |last=Johnson |url=http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 |title=A Treatise on Ordinary and Partial Differential Equations |publisher=John Wiley and Sons |year=1913 }} In [http://hti.umich.edu/u/umhistmath/ University of Michigan Historical Math Collection]
* {{cite book |first1=A. D. |last1=Polyanin |first2=V. F. |last2=Zaitsev |title=Handbook of Exact Solutions for Ordinary Differential Equations |edition=2nd |publisher=Chapman & Hall/CRC Press |location=Boca Raton |year=2003 |isbn=1-58488-297-2 }}
* {{cite book |first=R. I. |last=Porter |title=Further Elementary Analysis |year=1978 |chapter=XIX Differential Equations }}
* {{Cite book| last = Teschl| given = Gerald|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* {{cite book|author=Daniel Zwillinger|title=Handbook of Differential Equations|url=https://books.google.com/books?id=n7TiBQAAQBAJ&q=%22Handbook+of+Differential+Equations%22|date=12 May 2014|publisher=Elsevier Science|isbn=978-1-4832-6396-0}}

==External links==
{{wikiquote}}
{{Wikibooks|Ordinary Differential Equations}}
{{Wikiversity|Differential equations}}
{{EB1911 poster|Differential Equation}}
* {{Commons category-inline|Differential equations}}
* [http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/ Lectures on Differential Equations] [[MIT]] Open CourseWare Videos
* [http://tutorial.math.lamar.edu/classes/de/de.aspx Online Notes / Differential Equations] Paul Dawkins, [[Lamar University]]
* [http://www.sosmath.com/diffeq/diffeq.html Differential Equations], S.O.S. Mathematics
* [http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm Introduction to modeling via differential equations] Introduction to modeling by means of differential equations, with critical remarks.
* [http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode Mathematical Assistant on Web] Symbolic ODE tool, using [[Maxima (software)|Maxima]]
* [http://eqworld.ipmnet.ru/en/solutions/ode.htm Exact Solutions of Ordinary Differential Equations]
* [http://www.hedengren.net/research/models.htm Collection of ODE and DAE models of physical systems] {{Webarchive|url=https://web.archive.org/web/20081219003453/http://www.hedengren.net/research/models.htm |date=2008-12-19 }} MATLAB models
* [http://www.jirka.org/diffyqs/ Notes on Diffy Qs: Differential Equations for Engineers] An introductory textbook on differential equations by Jiri Lebl of [[UIUC]]
* [http://www.khanacademy.org/math/differential-equations Khan Academy Video playlist on differential equations ] Topics covered in a first year course in differential equations.
* [https://web.archive.org/web/20130607120716/http://math.rareinfos.com/category/courses/solutions-differential-equations/homogeneous-linear-systems/ MathDiscuss Video playlist on differential equations ]

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{{Differential equations topics}}
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[[Category:Differential equations| ]]