Differential equation (idea)

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,...,df_n) = 0
where F is a function F: Z x X x Y x D x ... x D -> Y^m, the D-parameters df_i 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:

• f(t) = f'(t) equivalent to f(t) - f'(t) = 0, the first form is called an explicit differential equation, this form doesn't always have to exist.

• df        df
  --(t,x) = --(t,x)
  dt        dx
  

• F'(A)H = - H^-1AH , where A,H invertible matrices, diff. eqn. on the space of matrices.
  Solution: F(A) = A^-1