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erential equation describing the motion of a pendulum of length l l d 2 u d x 2 g sin u 0 displaystyle l frac d 2 u dx 2 g sin u 0 the next group of examples are partial differential equations the unknown function u depends on two variables x and t or x and y homogeneous first order linear partial differential equation u t t u x 0 displaystyle frac partial u partial t t frac partial u partial x 0 homogeneous second order linear constant coefficient partial differential equation of elliptic type the laplace equation 2 u x 2 2 u y 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 0 third order non linear partial differential equation the kdv equation u t 6 u u x 3 u x 3 displaystyle frac partial u partial t 6u frac partial u partial x frac partial 3 u partial x 3 initial conditions and boundary conditions edit 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 a n displaystyle n th order ode contains n 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 of initial 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 an initial 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 a boundary value problem more generally the term initial conditions is normally used when the conditions are given at the same value of the independent variable and the term boundary 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 edit 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 the peano 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 region z displaystyle z such that z l m n p displaystyle z l m times n p and a b displaystyle a b is in the interior of z displaystyle z if we are given a differential equation d y d x g x y textstyle frac dy dx g x y and the condition that y b displaystyle y b when x a displaystyle x a then there is locally a solution to this problem if g x y displaystyle g x y is continuous on z displaystyle z this solution exists on some interval with its center at a displaystyle a the solution may not be unique see ordinary 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 f n x d n y d x n f 1 x d y d x f 0 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 x 0 y 0 y x 0 y 0 y x 0 y 0 displaystyle begin aligned y x_ 0 y_ 0 y x_ 0 y _ 0 y x_ 0 y _ 0 ldots end aligned for any nonzero f n x displaystyle f_ n x if f 0 f 1 displaystyle f_ 0 f_ 1 ldots and g displaystyle g are continuous on some interval containing x 0 displaystyle x_ 0 y displaystyle y exists and is unique 15 related concepts edit 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 equations may be viewed as the analog to differential equations where instead of the equation involving derivatives the equation contains integrals 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 edit differential equations are closely related to difference 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 the euler method involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation applications edit the study of differential equations is a wide field in 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 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 have closed form solutions instead solutions can be approximated using numerical methods many fundamental laws of physics and chemistry can be formulated as differential equations in biology and economics differential equations are used to model 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 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 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 number of differential equations that have received a name in various scientific areas demonstrates the importance of the topic see list of named differential equations software edit some cas software can solve differential equations these are the commands used in the leading programs maple 16 dsolve mathematica 17 dsolve maxima 18 ode2 equation y x sagemath 19 desolve sympy 20 sympy solvers ode dsolve equation xcas 21 desolve y k y y see also edit 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 references edit dennis g zill 15 march 2012 a first course in differential equations with modeling applications cengage learning isbn 978 1 285 40110 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 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 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 frasier craig july 1983 review of the 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 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 for a special collection of the 9 groundbreaking papers by the three authors see first appearance of the wave equation d alembert leonhard euler daniel bernoulli the controversy about vibrating strings archived 2020 02 09 at the wayback machine retrieved 13 nov 2012 herman hj lynge and son for lagrange s contributions to the acoustic wave equation consult acoustics an introduction to its physical principles and applications allan d pierce acoustical soc of america 1989 page 18 retrieved 9 dec 2012 speiser david discovering the principles of mechanics 1600 1800 p 191 basel birkhäuser 2008 fourier joseph 1822 théorie analytique de la chaleur in french paris firmin didot père et fils oclc 2688081 boyce william e diprima richard c 1967 elementary differential equations and boundary value problems 4th ed john wiley sons p 3 weisstein eric w ordinary differential equation order from mathworld a wolfram web resource http mathworld wolfram com ordinarydifferentialequationorder html order and degree of a differential equation accessed dec 2015 elias loomis 1887 elements of the differential and integral calculus revised ed harper bros p 247 extract of page 247 zill dennis g 2001 a first course in differential equations 5th ed brooks cole isbn 0 534 37388 7 dsolve maple programming help www maplesoft com retrieved 2020 05 09 dsolve wolfram language documentation www wolfram com retrieved 2020 06 28 schelter william f gaertner boris ed differential equations symbolic solutions the computer algebra program maxima a tutorial in maxima documentation on sourceforge archived from the original on 2022 10 04 basic algebra and calculus sage tutorial v9 0 doc sagemath org retrieved 2020 05 09 ode sympy 1 11 documentation 2022 08 22 archived from the original on 2022 09 26 symbolic algebra and mathematics with xcas pdf further reading edit abbott p neill hugh 2003 calculus london chicago hodder stoughton educational contemporary books pp 266 277 isbn 978 0 07 142128 7 blanchard paul 2006 student solutions manual for blanchard devaney and hall s differential equations 3rd ed belmont ca brooks cole thomson learning isbn 978 0 495 01461 4 boyce william e diprima richard c meade douglas b 2017 elementary differential equations and boundary value problems 11th ed hoboken nj john wiley sons isbn 978 1 119 44376 6 coddington e a levinson n 1955 theory of ordinary differential equations new york mcgraw hill ince e l 1956 ordinary differential equations new york dover johnson w 1913 a treatise on ordinary and partial differential equations john wiley and sons in university of michigan historical math collection polyanin a d zaitsev v f 2003 handbook of exact solutions for ordinary differential equations 2nd ed boca raton chapman hall crc press isbn 1 58488 297 2 porter ronald i 1978 xix differential equations further elementary analysis 4th ed london bell hyman isbn 978 0 7135 1594 7 teschl gerald 2012 ordinary differential equations and dynamical systems providence ri american mathematical society isbn 978 0 8218 8328 0 zwillinger daniel 2014 handbook of differential equations 2nd ed burlington elsevier science isbn 978 1 4832 6396 0 external links edit wikiquote has quotations related to differential equation wikibooks has a book on the topic of ordinary differential equations wikiversity has learning resources about differential equations wikisource has the text of the 1911 encyclopædia britannica article differential equation media related to differential equations at wikimedia commons lectures on differential equations mit open courseware videos online notes differential equations paul dawkins lamar university differential equations s o s mathematics introduction to modeling via differential equations introduction to modeling by means of differential equations with critical remarks mathematical assistant on web symbolic ode tool using maxima exact solutions of ordinary differential equations collection of ode and dae models of physical systems archived 2008 12 19 at the wayback machine matlab models notes on diffy qs differential equations for engineers an introductory textbook on differential equations by jiri lebl of uiuc khan academy video playlist on differential equations topics covered in a first year course in differential equations mathdiscuss video playlist on differential equations v t e major mathematics areas history timeline future lists glossary foundations category theory information theory mathematical logic order theory philosophy of mathematics set theory type theory algebra abstract commutative elementary field theory group theory homological linear multilinear analysis calculus real analysis complex analysis differential equations functional analysis harmonic analysis measure theory discrete combinatorics discrete geometry graph theory geometry algebraic numerical algebraic affine analytic arithmetic complex computational convex differential discrete euclidean finite information projective synthetic trigonometry number theory algebraic analytic arithmetic diophantine geometry topology general arithmetic algebraic differential geometric homotopy theory applied engineering mathematics ethnomathematics mathematical biology mathematical chemistry mathematical economics mathematical finance mathematical geology mathematical linguistics mathematical physics mathematical psychology mathematical sociology mathematical statistics probability statistics systems science control theory game theory operations research computational computer science theory of computation 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