A differential equation, is an equation where there exist variables for a function f: X -> Y ( X,Y Banach spaces), and its derivates, both ordinary and partial allowed.
Let X,Y be Banach spaces, Z the set of functions X -> Y, D the set of derivates, both partial and ordinary allowed, which do not have to be defined on the whole X.
An (implicit) differential equation is an equation
F(f,x,df1,...,dfn) = 0
where F is a function F: Z x X x Y x D x ... x D -> Ym, the D-parameters dfi must be exactly same type of derivative for a fixed i, m > 0 an integer.
A function f: X -> Y is called a solution iff F applied to f and its derivatives needed for F is 0 for an non-empty (open ?) subset of X x Y.
This definition is quite ugly, but should capture all types of differential equations like ordinary differential equations or partial differential equations.
Examples: