A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line to any desired level of accuracy.
Precisely, a (possibly implicit) differential equation 
is a UDE if for any continuous real-valued function 
and for any positive continuous function 
there exist a smooth solution 
of with 
for all 
.[1]
The existence of an UDE has been initially regarded as an analogue of the universal Turing machine for analog computers, because of a result of Shannon that identifies the outputs of the general purpose analog computer with the solutions of algebraic differential equations.[1] However, in contrast to universal Turing machines, UDEs do not dictate the evolution of a system, but rather sets out certain conditions that any evolution must fulfill.[2]
- Rubel found the first known UDE in 1981. It is given by the following implicit differential equation of fourth-order:[1][2]
- Duffin obtained a family of UDEs given by:[3]
and 
, whose solutions are of class 
for n > 3.
, where n > 3.
- Bournez and Pouly proved the existence of a fixed polynomial vector field p such that for any f and ε there exists some initial condition of the differential equation y' = p(y) that yields a unique and analytic solution satisfying |y(x) − f(x)| < ε(x) for all x in R.[2]
