34: Ordinary differential equations
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.
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).
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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.
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