Differential equation
Not to be confused with Difference equation.
ter 2 of his 1671 work “Methodus fluxionum et Serierum
A differential equation is a mathematical equation Infinitarum”,[1] Isaac Newton listed three kinds of differential equations:
dy
= f (x)
dx
dy
= f (x, y)
dx
∂y
∂y
x1
+ x2
=y
∂x1
∂x2
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.[2] This is an ordinary differential equation
of the form
Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the
casing and being cooled at the boundary, providing a steady state
temperature distribution.
y ′ + P (x)y = Q(x)y n
from several different perspectives, mostly concerned
with their solutions—the set of functions that satisfy the
equation. Only the simplest differential equations are
solvable by explicit formulas; however, some properties
of solutions of a given differential equation may be determined without finding their exact form.
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.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using
computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have
been developed to determine solutions with a given degree of accuracy.
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.
for which the following year Leibniz obtained solutions
[3]
that relates some function with its derivatives. In applica- by simplifying it.
tions, the functions usually represent physical quantities, Historically, the problem of a vibrating string such as
the derivatives represent their rates of change, and the that of a musical instrument was studied by Jean le
equation defines a relationship between the two. Because Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and
such relations are extremely common, differential equa- Joseph-Louis Lagrange.[4][5][6][7] In 1746, d’Alembert
tions play a prominent role in many disciplines including discovered the one-dimensional wave equation, and
engineering, physics, economics, and biology.
within ten years Euler discovered the three-dimensional
[8]
In pure mathematics, differential equations are studied wave equation.
Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat),[9] 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 differ1 History
ence of their temperatures. Contained in this book was
Fourier’s proposal of his heat equation for conductive difDifferential equations first came into existence with the fusion of heat. This partial differential equation is now
invention of calculus by Newton and Leibniz. In Chap- taught to every student of mathematical physics.
1
2
2
3
Example
TYPES
functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical
and numerical methods, applied by hand or by computer,
may approximate solutions of ODEs and perhaps yield
useful information, often sufficing in the absence of exact, analytic solutions.
For example, in classical mechanics, the motion of a body
is described by its position and velocity as the time value
varies. Newton’s laws allow (given the position, velocity,
acceleration and various forces acting on the body) one to
express these variables dynamically as a differential equation for the unknown position of the body as a function
3.2
of time.
Partial differential equations
In some cases, this differential equation (called an Main article: Partial differential equation
equation of motion) may be solved explicitly.
An example of modelling 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 acceleration
due to air resistance.
A partial differential equation (PDE) is a differential
equation that contains unknown multivariable functions
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 variGravity is considered constant, and air resistance may be ables, and are either solved by hand, or used to create a
modeled as proportional to the ball’s velocity. This means relevant computer model.
that the ball’s acceleration, which is a derivative of its ve- PDEs can be used to describe a wide variety of phenomlocity, depends on the velocity (and the velocity depends ena such as sound, heat, electrostatics, electrodynamics,
on time). Finding the velocity as a function of time in- fluid flow, elasticity, or quantum mechanics. These
volves solving a differential equation and verifying its va- seemingly distinct physical phenomena can be forlidity.
malised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional
dynamical systems, partial differential equations often
model multidimensional systems. PDEs find their gen3 Types
eralisation in stochastic partial differential equations.
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/Partial, Linear/Non-linear, and Homogeneous/Inhomogeneous. This list is far from exhaustive;
there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
3.1
Ordinary differential equations
Main article: Ordinary differential equation
An ordinary differential equation (ODE) is an equation
containing a function of one independent variable and its
derivatives. The term "ordinary" is used in contrast with
the term partial differential equation which may be with
respect to more than one independent variable.
3.3 Linear differential equations
Main article: Linear differential equation
A differential equation is linear if the unknown function
and its derivatives appear to the power 1 (products of the
unknown function and its derivatives are not allowed) and
nonlinear otherwise. The characteristic property of linear
equations is that their solutions form an affine subspace
of an appropriate function space, which results in much
more developed theory of linear differential equations.
Homogeneous linear differential equations are a subclass
of linear differential equations for which the space of
solutions is a linear subspace i.e. the sum of any set
of solutions or multiples of solutions is also a solution.
The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be
(known) functions of the independent variable or variables; if these coefficients are constants then one speaks
of a constant coefficient linear differential equation.
Linear differential equations, which have solutions that
can be added and multiplied by coefficients, are welldefined and understood, and exact closed-form solutions 3.4 Non-linear differential equations
are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more in- There are very few methods of solving nonlinear differtricate, as one can rarely represent them by elementary ential equations exactly; those that are known typically
3
depend on the equation having particular symmetries.
Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of 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.[10]
Linear differential equations frequently appear as
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 (see below).
3.5
Equation order
d2 u
+ ω 2 u = 0.
dx2
• Inhomogeneous first-order nonlinear ordinary differential equation:
du
= u2 + 4.
dx
• Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of
a pendulum of length L:
L
d2 u
+ g sin u = 0.
dx2
In the next group of examples, the unknown function u
depends on two variables x and t or x and y.
Differential equations are described by their order, determined by the term with highest number of derivatives.
An equation containing only single derivatives is a firstorder differential equation, an equation containing double
derivatives is a second-order differential equation, and so
on.[11][12]
• Homogeneous first-order linear partial differential
equation:
3.6
• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the
Laplace equation:
Examples
In the first group of examples, let u be an unknown function of x, and c and ω are known constants. Note both ordinary and partial differential equations are broadly classified as linear and nonlinear.
• Inhomogeneous first-order linear constant coefficient ordinary differential equation:
du
= cu + x2 .
dx
• Homogeneous second-order linear ordinary differential equation:
du
d2 u
−x
+ u = 0.
2
dx
dx
∂u
∂u
+t
= 0.
∂t
∂x
∂2u ∂2u
+ 2 = 0.
∂x2
∂y
• Third-order nonlinear partial differential equation,
the Korteweg–de Vries equation:
∂u ∂ 3 u
∂u
= 6u
−
.
∂t
∂x ∂x3
4 Existence of solutions
Solving differential equations is not like solving algebraic
equations. Not only are their solutions oftentimes unclear, but whether solutions are unique or exist at all are
also notable subjects of interest.
• Homogeneous second-order linear constant coeffi- For first order initial value problems, the Peano existence
cient ordinary differential equation describing the theorem gives one set of circumstances in which a soluharmonic oscillator:
tion exists. Given any point (a, b) in the xy-plane, define
4
7 APPLICATIONS
some rectangular region Z , such that Z = [l, m] × [n, p]
and (a, b) is in the interior of Z . If we are given a differdy
ential equation dx
= g(x, y) and the condition that y = b
when x = a , then there is locally a solution to this prob∂g
lem if g(x, y) and ∂x
are both continuous on Z . This
solution exists on some interval with its center at a . The
solution may not be unique. (See Ordinary differential
equation for other results.)
7 Applications
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 emphasizes the rigorous justiHowever, this only helps us with first order initial value fication of the methods for approximating solutions. Difproblems. Suppose we had a linear initial value problem ferential equations play an important role in modelling
virtually every physical, technical, or biological process,
of the nth order:
from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used
dn y
dy
to solve real-life problems may not necessarily be directly
fn (x) n + · · · + f1 (x)
+ f0 (x)y = g(x)
dx
dx
solvable, i.e. do not have closed form solutions. Instead,
solutions can be approximated using numerical methods.
such that
y(x0 ) = y0 , y ′ (x0 ) = y0′ , y ′′ (x0 ) = y0′′ , · · ·
For any nonzero fn (x) , if {f0 , f1 , · · · } and g are continuous on some interval containing x0 , y is unique and
exists.[13]
5
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.
• 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 differential algebraic equation (DAE) is a differential equation comprising differential and algebraic
terms, given in implicit form.
6
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 approximation of the solution of a differential equation by the solution of a corresponding difference equation.
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.
Whenever this happens, mathematical theory behind the
equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider 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.
7.1 Physics
• Euler–Lagrange equation in classical mechanics
• Hamilton’s equations in classical mechanics
• Radioactive decay in nuclear physics
• Newton’s law of cooling in thermodynamics
• The wave equation
• The heat equation in thermodynamics
• Laplace’s equation, which defines harmonic functions
• Poisson’s equation
7.2
Biology
• The geodesic equation
5
7.1.4 Quantum mechanics
• The Navier–Stokes equations in fluid dynamics
In quantum mechanics, the analogue of Newton’s law is
Schrödinger’s equation (a partial differential equation)
• The Diffusion equation in stochastic processes
for a quantum system (usually atoms, molecules, and
subatomic particles whether free, bound, or localized).
• The Convection–diffusion equation in fluid dynam- It is not a simple algebraic equation, but in general a
ics
linear partial differential equation, describing the timeevolution of the system’s wave function (also called a
• The Cauchy–Riemann equations in complex analy- “state function”).[17]:1–2
sis
• The Poisson–Boltzmann equation in molecular dy- 7.2 Biology
namics
• Verhulst equation – biological population growth
• The shallow water equations
• von Bertalanffy model – biological individual growth
• Universal differential equation
• Replicator dynamics – found in theoretical biology
• The Lorenz equations whose solutions exhibit
chaotic flow.
• Hodgkin–Huxley model – neural action potentials
7.2.1 Predator-prey equations
7.1.1
Classical mechanics
So long as the force acting on a particle is known,
Newton’s second law is sufficient to describe the motion
of a particle. Once independent relations for each force
acting on a particle are available, they can be substituted
into Newton’s second law to obtain an ordinary differential equation, which is called the equation of motion.
7.1.2
Electrodynamics
The Lotka–Volterra equations, also known as the
predator–prey equations, are a pair of first-order, nonlinear, differential equations frequently used to describe
the dynamics of biological systems in which two species
interact, one as a predator and the other as prey.
7.3 Chemistry
The rate law or rate equation for a chemical reaction is
a differential equation that links the reaction rate with
concentrations or pressures of reactants and constant parameters (normally rate coefficients and partial reaction
orders).[18] To determine the rate equation for a particular system one combines the reaction rate with a mass
balance for the system.[19]
Maxwell’s equations are a set of partial differential equations that, together with the Lorentz force law, form the
foundation of classical electrodynamics, classical optics,
and electric circuits. These fields in turn underlie modern
electrical and communications technologies. Maxwell’s
equations describe how electric and magnetic fields are
generated and altered by each other and by charges and 7.4 Economics
currents. They are named after the Scottish physicist and
mathematician James Clerk Maxwell, who published an
• The key equation of the Solow–Swan model is
∂k(t)
α
early form of those equations between 1861 and 1862.
∂t = s[k(t)] − δk(t)
• The Black–Scholes PDE
7.1.3
General relativity
The Einstein field equations (EFE; also known as “Einstein’s equations”) are a set of ten partial differential
equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of
gravitation as a result of spacetime being curved by matter
and energy.[14] First published by Einstein in 1915[15]
as a tensor equation, the EFE equate local spacetime
curvature (expressed by the Einstein tensor) with the local
energy and momentum within that spacetime (expressed
by the stress–energy tensor).[16]
• Malthusian growth model
• The Vidale–Wolfe advertising model
8 See also
• Complex differential equation
• Exact differential equation
• Initial condition
6
10 FURTHER READING
• Integral equations
• Numerical methods
• Picard–Lindelöf theorem on existence and uniqueness of solutions
• Recurrence relation, also known as 'Difference
Equation'
9
References
[1] 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].
[2] 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
[3] Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard
(1993), Solving ordinary differential equations I: Nonstiff
problems, Berlin, New York: Springer-Verlag, ISBN 9783-540-56670-0
[4] Cannon, John T.; Dostrovsky, Sigalia (1981). “The evolution of dynamics, vibration theory from 1687 to 1742”.
Studies in the History of Mathematics and Physical Sciences 6. New York: Springer-Verlag: ix + 184 pp. ISBN
0-3879-0626-6. GRAY, JW (July 1983). “BOOK REVIEWS”. BULLETIN (New Series) OF THE AMERICAN
MATHEMATICAL SOCIETY 9 (1). (retrieved 13 Nov
2012).
[5] 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.
[6] 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 (retrieved 13 Nov
2012). Herman HJ Lynge and Son.
[7] For de Lagrange’s contributions to the acoustic wave equation, can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
[8] Speiser, David. Discovering the Principles of Mechanics
1600-1800, p. 191 (Basel: Birkhäuser, 2008).
[9] Fourier, Joseph (1822). Théorie analytique de la chaleur
(in French). Paris: Firmin Didot Père et Fils. OCLC
2688081.
[10] Boyce, William E.; DiPrima, Richard C. (1967). Elementary Differential Equations and Boundary Value Problems
(4th ed.). John Wiley & Sons. p. 3.
[11] Weisstein, Eric W. “Ordinary Differential Equation
Order.”
From
MathWorld--A
Wolfram
Web Resource.
http://mathworld.wolfram.com/
OrdinaryDifferentialEquationOrder.html
[12] Order and degree of a differential equation, accessed Dec
2015.
[13] Zill, Dennis G. A First Course in Differential Equations
(5th ed.). Brooks/Cole. ISBN 0-534-37388-7.
[14] Einstein, Albert (1916).
“The Foundation of the
General Theory of Relativity” (PDF). Annalen der
Physik 354 (7): 769. Bibcode:1916AnP...354..769E.
doi:10.1002/andp.19163540702.
[15] Einstein, Albert (November 25, 1915). “Die Feldgleichungen der Gravitation”. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847.
Retrieved 2006-09-12.
[16] Misner, Charles W.; Thorne, Kip S.; Wheeler, John
Archibald (1973). Gravitation. San Francisco: W. H.
Freeman. ISBN 978-0-7167-0344-0 Chapter 34, p. 916.
[17] Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7
[18] IUPAC Gold Book definition of rate law. See also: According to IUPAC Compendium of Chemical Terminology.
[19] Kenneth A. Connors Chemical Kinetics, the study of reaction rates in solution, 1991, VCH Publishers.
10 Further reading
• P. Abbott and H. Neill, Teach Yourself Calculus,
2003 pages 266-277
• P. Blanchard, R. L. Devaney, G. R. Hall, Differential
Equations, Thompson, 2006
• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955
• E. L. Ince, Ordinary Differential Equations, Dover
Publications, 1956
• W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in
University of Michigan Historical Math Collection
• A. D. Polyanin and V. F. Zaitsev, Handbook of
Exact Solutions for Ordinary Differential Equations
(2nd edition), Chapman & Hall/CRC Press, Boca
Raton, 2003. ISBN 1-58488-297-2.
• R. I. Porter, Further Elementary Analysis, 1978,
chapter XIX Differential Equations
• Teschl, Gerald (2012).
Ordinary Differential
Equations and Dynamical Systems. Providence:
American Mathematical Society. ISBN 978-08218-8328-0.
7
• D. Zwillinger, Handbook of Differential Equations
(3rd edition), Academic Press, Boston, 1997.
11
External links
• 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 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
8
12
12
12.1
TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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