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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 34: Ordinary differential equations

Introduction

Ordinary differential equations are equations to be solved in which the unknown element is a function, rather than a number, and in which the known information relates that function to its derivatives. Few such equations admit an explicit answer, but there is a wealth of qualitative information describing the solutions and their dependence on the defining equation. There are many important classes of differential equations for which detailed information is available. Applications to engineering and the sciences abound. Numerical solutions are actively studied.

History

Applications and related fields

Note that every indefinite integration problem is really an example of a differential equation, so the entirety of section 28: Integration and Measure is subsumed in this section in principle.

The solutions to many classic differential equations, particularly linear second-order differential equations, cannot be expressed in terms of the elementary functions but are themselves studied in 33: Special Functions. This includes Bessel functions, Whittaker functions, Airy functions, and so on.

A comparatively small collection of differential equations can be solved in "closed form" using simple functions. The tools of 12H: Differential and difference algebra can be used to identify these.

Stochastic differential equations are part of 60: Probability Theory

Numerical solutions of differential equations is a branch of Numerical Analysis.

For applications of differential equations, see the sections 70 through 86 of the MSC (classical applications of mathematics to the physical sciences).

This image slightly hand-edited for clarity.

Subfields

• 34A: General theory
• 34B: Boundary value problems, For ordinary differential operators, see 34LXX
• 34C: Qualitative theory, see also 58FXX
• 34D: Stability theory, see also 58F10, 93DXX
• 34E: Asymptotic theory
• 34F05: Equations and systems with randomness, See also 34K50, 60H10, 93E03
• 34G: Differential equations in abstract spaces, see also 58D25
• 34H05: Control problems, See also 49J25, 49K25, 93C15
• 34K: Functional-differential and differential-difference equations with or without deviating arguments
• 34L: Ordinary differential operators, see also 47E05
• 34M: Differential equations in the complex domain (See also 30Dxx, 32G34) [new in 2000]

This is one of the larger areas of the Math Reviews database.

Through 1958 there was an additional subject heading, "37: Differential equations, functional calculus".

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

There is a bumper crop of texts available at the undergraduate level; we decline to single out any one at this level. Typical topics seem to include various special classes of functions admitting (nearly) closed-form solutions (first order linear, linear with constant coefficients, separable, etc.); general tools (Laplace transforms, variation of parameters); some numerical methods (Euler's method, Runge-Kutta); and a few existence and uniqueness theorems.

A somewhat more advanced undergraduate text is O'Malley, Robert E., Jr.: "Thinking about ordinary differential equations", Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1997. 247 pp. ISBN 0-521-55314-8; 0-521-55742-9 MR98c:34002

A graduate text with a good broad view is Kartsatos, Athanassios G.: "Advanced ordinary differential equations", Mariner Publishing Co., Inc., Tampa, Fla., 1980. 186 pp., ISBN 0-936166-02-9, MR83d:34004

Rouche, N.; Mawhin, J.: "Ordinary differential equations. Stability and periodic solutions", Surveys and Reference Works in Mathematics, 5. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. 260 pp. ISBN 0-273-08419-4 MR82i:34001 (French original, with companion first volume: MR 58#1318ab)

Rassias, John M.: "Counterexamples in differential equations and related topics", World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. 184 pp. ISBN 981-02-0460-4

• Links to online course materials by several authors.
• Online text by Christopher A. Barker
• Online text by Angus MacKinnon

Software and tables

• Ordinary differential equations software
• Differential Equations Resources
• Differential-equations-by-mail server
• Java applet allowing graphic solutions of autonomous systems of two dependent variables.

Other web sites with this focus

• SIAM's Differential Equations page.
• UTK archives page
• Here are the AMS and Goettingen resource pages for area 34.

Selected topics at this site

• Review of terms for classifying first-order ODEs.
• Proofs of existence, uniqueness of solutions of initial value ODE problems
• First-order linear differential equation with smooth, nonanalytic solution
• Basic tricks for solving second-order autonomous ODEs
• Most ODEs do not admit an explicit solution expressible using elementary functions. Determining when such solutions do exist is a branch of symbolic algebra; see e.g. this discussion of symbolic integration. (Includes references for symbolic solutions of ODEs).
• Solve this first order ODE: y'=a/y+by/x+c/sqrt(x).
• Solving f'=f o f (or, "What to Ask When Asking About Differential Equations")
• Discussion of solutions to a differential equation arising in economics. Discussion includes being careful about technical requirements assumed.
• When did it start to snow?
• Integrating the solution to an ODE x'=f(x).
• How closely do asymptotics of the coefficients of a system of differential equation mirror the asymptotics of the solution? (not necessarily closely).
• Using the asymptotics on an ODE to determine the asymptotic behaviour of its solutions.
• System of DEs which model highly oscillatory motion on a sphere.
• Rolling balls: a nearly-linear system of 3 first-order differential equations.
• Analyzing a system of three first-order ODEs.
• A 2nd order linear equation. After the fact we have learned (a)about Bessel functions (2)to use a symbolic algebra program to solve differential equations.
• Find the differential equation satisfied by a family of functions
• Abelian integrals: y"=k(y^2). Bonus Offer: article includes careful distinct between variables and functions. How to handle 2nd order ODE with no y' term.
• Autonomous system of two linear differential equations.
• Describing solutions of autonomous systems of differential equations.
• Computing the solution curves of predator-prey models (autonomous systems of 2 differential equations.)
• Who says you can't get rich solving a system of ODEs?
• Estimating integrals of solutions to a differential equation.
• Eliminating variables from systems of ODEs using resultants
• How do Lie groups enter the analysis of a differential equation?
• Applications of Lie theory for solving differential equations
• How do Lie groups and algebras fit in among the branches of mathematics?
• Solving linear differential equations with linear coefficients.
• Concerning linear differential equations with polynomial coefficients.
• A simple delay differential equation
• Differential Algebraic Equations (DAEs) -- systems with both differential equations and algebraic equations
• Mixed differential-difference equations (differential and functional equations)
• Exploiting the parallels between differential equations and difference equations