A
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
Every ODE can be transformed in an autonomous ODE, but this doesn't usually help much.