The central piece of Rungu-Kutta method is the approximation of the increasement of the function. In 1st order ODE,

In a special case of Rungu-Kutta of order 4 (RK4), there are 2 array and , so that

where is ranging from 0 to 1, .

In RK4,

The is a must, otherwise, we have to define . There should be some methods to obtain an optimum values for and , but I don’t know.

For 2nd order ODE

change to

These equation are the similar 1st order ODEs.

, where is the z for step.

The is using

This can be generalized to any order ODE by decoupling the ODE into

the equation for is

And for all the intermediate variable

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