Among the most important numerical methods are the Runge-Kutta methods. In this exercise, we will use one of these methods to estimate the solution to a differential equation. Specifically, let's say that the differential equation can be written in the form y' = f(t, y) That is, the derivative of y is a function of both time t and the actual function y itself. We also assume that we know the initial condition y (to ) = yo . Our objective is to approximate the value of y at time t = to . We select a step-size of time, which we'll represent by h. Starting at time t = 0, our strategy will be to move forward one step-size at a time, solving for the approximate value of the function y (t) at that time step, then using that mo approximation as the basis for taking our next time step. We will continue that process until we reach time to , at which point we'll stop. To state the algorithm in its most general form, we define time step n+1 as tn+1 = tn + h Knowing the approximate solution yn at time step tn , we find the approximate solution Un+1 at time tn+1 using the formulas Un+1 = Un + 6 (k1 + 2k2 + 2k3 + K4) where k1 = f (tn, yn) k:2 = f (tn + h, Un + 62 ) ks = f (tn + ", Un + h2 ) KA = f (tn + h, yn + k3) We continue this iterative loop until we reach our desired endpoint in time to . Stated in pseudocode: n = 0 to = 0 y (to) = yo WHILE tn