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y differential equation ode is an equation containing an unknown function of one real or complex variable x its derivatives and some given functions of x the unknown function is generally represented by a dependent variable often denoted y which therefore depends on x thus 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 as in general the solutions of a differential equation cannot be expressed by a closed form expression numerical methods are commonly used for solving differential equations on a computer partial differential equations edit main article partial differential equation a partial differential equation pde is a differential equation that contains unknown multivariable functions and their partial derivatives pdes are used to formulate problems involving functions of several variables and are either solved in closed form or using 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 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 linear differential equations edit main article linear differential equations linear differential equations are differential equations that are 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 integrals many differential equations that are encountered in physics are linear for example odes describing radioactive decay and pdes for heat transfer by thermal diffusion these lead to special functions which may be defined as solutions of linear differential equations see holonomic function non linear differential equations edit main article 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 symmetries nonlinear differential equations can exhibit very complicated behaviour over extended time intervals characteristic of chaos even the fundamental questions of existence and uniqueness of solutions for nonlinear differential equations are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory cf navier stokes existence and smoothness however if the differential equation is a correctly formulated representation of a meaningful physical process then one expects it to have a solution 11 in some circumstances nonlinear differential equations may be approximated by linear ones 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 similarly when a fixed point or stationary solution of a nonlinear differential equation has been found investigation of its stability leads to a linear differential equation equation order and degree edit 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 derivatives is a first order differential equation an equation containing the second order derivative is a second order differential equation and so on 12 13 when it is written as a polynomial equation in the unknown function and its derivatives the degree of the differential equation is depending on the context the polynomial degree in the highest derivative of the unknown function 14 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 y y 2 0 displaystyle y y 2 0 is of degree one for the first meaning but not for the second one differential equations that describe natural phenomena usually have only first and second order derivatives in them but there are some exceptions such as the thin film equation which is a fourth order partial differential equation homogeneous linear equations edit a linear differential equation is homogeneous if each term in the equation includes either the dependent variable or one of its derivatives if this is not the case so that there is a term that does not include either the dependent variable itself or a derivative of it the equation is inhomogeneous or heterogeneous see the examples section below examples edit the first group of examples are ordinary differential equations where u is an unknown function of x and c and ω are constants that are assumed to be known these examples illustrate the distinction between linear and nonlinear differential equations and between homogeneous differential equations and inhomogeneous ones defined above inhomogeneous first order linear constant coefficient ordinary differential equation d u d x c u x 2 displaystyle frac du dx cu x 2 homogeneous second order linear ordinary differential equation d 2 u d x 2 x d u d x u 0 displaystyle frac d 2 u dx 2 x frac du dx u 0 homogeneous second order linear constant coefficient ordinary differential equation describing the harmonic oscillator d 2 u d x 2 ω 2 u 0 displaystyle frac d 2 u dx 2 omega 2 u 0 first order nonlinear ordinary differential equation d u d x u 2 4 displaystyle frac du dx u 2 4 second order nonlinear due to sine function ordinary differential 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 nørset...
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