Universal Differential Equation

A universal differential equation (UDE) is a fixed, nontrivial algebraic differential equation whose real solutions approximate every real-valued continuous function on the real line within any prescribed positive continuous function as an error bound.

Rubel (1981) found the first known UDE by showing that, given any continuous function and any positive continuous function , there exists a smooth solution of

(1)

such that

(2)

for all .

The construction uses the bump function

(3)

Its integral gives transition pieces whose derivatives of every positive order vanish at their endpoints. Translating and scaling these pieces and joining them gives the approximating smooth functions. Flat pieces also permit different solutions with the same initial value and derivatives, so Rubel's equation does not impose initial value problem uniqueness (Trott 2006; Bournez and Pouly 2020, pp. 28:2-28:3).

Duffin (1981) found two additional families of UDEs,

(4)

and

(5)

which admit approximating solutions of class for integer (Briggs 2002, p. 2).

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

(6)

for integer .

Bournez and Pouly (2020, p. 28:5) proved that there exists a fixed polynomial system with rational coefficients for which universality holds with unique analytic solutions. For any continuous function and positive continuous function , initial conditions can be chosen so that the unique solution satisfies for all .