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 
and any positive continuous function 
, there exists a 
solution 
of
 |
(1)
|
such that
 |
(2)
|
for all 
.
Duffin (1981) found two additional families of UDEs,
 |
(3)
|
and
 |
(4)
|
whose solutions are 
for 
.
Briggs (2002) found a further family of UDEs given by
 |
(5)
|
for 
.
-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.