**ordinary differential equation**(

**ODE**) is a differential equation where no partial derivatives are allowed. (X,Y again the Banach spaces)

An ODE is called **linear** iff it is an linear equation.

An ODE is called **autonomous** iff it doesn't depend on X.

An ODE is called **homogenous** iff the X and Y parameters are always multiplied with an f or df parameter.

There exist a decent existence theory for these and they are usually easier to solve (in some cases).

Examples:

- Again f(x)=f'(x), this autonomous, because the equtions doesn't depend on X
- f'(x) = f(x)
^{2}+ x, not autonomous, not homogenous - f'(x) = x f(x), homogenous but not autonomous