Side benefit from Runge-Kutta methods for ODE

[Namir]

I have been looking at various methods to numerically solve ODEs (Ordinary Differential Equations). Among the various methods you can find (in Wikipedia) a family of Runge-Kutta methods. The 4th order Runge-Kutta method is perhaps the most popular. It has been used by HP in various calculator Stat Pacs.

The family of various Runge-Kutta methods solve ODE of:

dy/dx = f(x,y)

One interesting "side effect" is that if:

dy/dx = f(x)

Then you can obtain methods for numerical integration. The 4th order Runge-Kutta method, in this case, becomes Simpson's rule. The interesting windfall here is using the various variants of Runge-Kutta as additional methods for numerical integration. These variant methods are basically a "free bonus" method for numerical integration, which you can explore and use in performing numerical integration.

Namir