Universal Differential Equation

A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.

Rubel (1981) found the first known UDE by showing that, given any continuous function phi:R->R and any positive continuous function epsilon:R->R^+, there exists a C^infty solution y of


such that


for all t in R.

Duffin (1981) found two additional families of UDEs,




whose solutions are C^n for n>3.

Briggs (2002) found a further family of UDEs given by


for n>3.

See also

Differential-Algebraic Equation

This entry contributed by Keith Briggs

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Boshernitzan, M. "Universal Formulae and Universal Differential Equations." Ann. Math. 124, 273-291, 1986.Boshernitzan, M. and Rubel, L. A. "Coherent Families of Polynomials." Analysis 6, 339-389, 1985.Briggs, K. "Another Universal Differential Equation." 8 Nov 2002., R. J. "Rubel's Universal Differential Equation." Proc. Nat. Acad. Sci. USA 78, 4661-4662, 1981.Elsner, C. "On the Approximation of Continuous Functions by C^infty-Solutions of Third-Order Differential Equations." Math. Nachr. 157, 235-241, 1992.Elsner, C. "A Universal Functional Equation." Proc. Amer. Math. Soc. 127, 139-143, 1999.Rubel, L. A. "A Universal Differential Equation." Bull. Amer. Math. Soc. 4, 345-349, 1981.Rubel, L. A. "Some Research Problems About Algebraic Differential Equations." Trans. Amer. Math. Soc. 280, 43-52, 1983.Rubel, L. A. "Some Research Problems About Algebraic Differential Equations II." Illinois J. Math. 36, 659-680, 1992.Rubel, L. A. "Uniform Approximation by Rational Functions All of Which Satisfy the Same Algebraic Differential Equation." J. Approx. Th. 84, 123-128, 1996.

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Universal Differential Equation

Cite this as:

Briggs, Keith. "Universal Differential Equation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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