Algebraic Differential Equation

An algebraic differential equation is a differential equation expressed as a polynomial relation among the independent variable, the unknown function, and finitely many of its derivatives. In one independent variable, it has the form

where is a nonzero polynomial with real or complex coefficients (Rubel 1992, p. 449).

The polynomial condition concerns the equation, rather than requiring its solutions to be algebraic functions. For example, the exponential function satisfies . Differential algebra studies these relations algebraically.

The term differs from differential-algebraic equation, which describes an implicit differential system whose defining function need not be a polynomial. The notions can overlap. A universal differential equation is an algebraic differential equation whose solution family has a specified approximation property.