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The Runge-Kutta method for integrating an ODE dy/dx = f(x,y) is derived by assuming the general form
yn+1 = yn + ak1 + bk2 + ...
where
k1 = h f(xn, yn)
k2 = h f(xn+alpha*h, yn+beta*k1)
etc. and h is the step size
and substituting in Taylor expansions to find coefficients (a, b, ..., alpha, beta, ...) that will cancel out error terms.

The most popular RK method is the 4th order Runge Kutta where

yn+1 = yn + (k1 + 2k2 + 2k3 + k4)/6
k1 = h f(xn, yn)
k2 = h f(xn+h/2, yn+k1/2)
k3 = h f(xn+h/2, yn+k2/2)
k4 = h f(xn+h, yn+k3)
The 4th order Runge-Kutta has local truncation error of order O(h5), giving total error of order O(h4). It is fairly stable, self-priming, and simple to implement, making it popular for quick-n-dirty applications. Generally Runge-Kutta is used with an adaptive step size algorithm to control the error.

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