When we use the average of the estimates of the derivatives at the end points, we can improve the approximation to the solution. A class of approximation techniques that estimates derivatives at various points within an interval and then computes a weighted average is the Runge-Kutta method, named for two German mathematicians. The RungeKutta methods are classified by order, where the order depends on the number of slope estimates used at each step. A very popular method is the fourth order Runge-Kutta method. Let:

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Yn+1 = yn + K + 2K + 2K3 + K4 6 K = g(tn. Yn)h K = g(n+h/2, yn + K/2)h K3 = g(tn +h/2, yn + K/2)h K = g(n+h, yn + K3)h + where Solve y' = 0.25 ty, y(0) = 2 with the fourth-order Runge-Kutta method using step sizes of 1.0, 0.5, and 0.1.