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Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy/dx = x-2.5y; y(0) = 4. Let f(x,y) = x-2.5y. We let x0 = 0 and y0 = 4 and pick a step size h = 0.25. The improved Euler method is the the following algorithm. From (xn,yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step x n+1 = xn + h, and then k1, the slope at (xn,yn). The predicted new value of the solution is zn+1 = yn + h k1. Then we find the slope at the predicted new point k2 = f(xn+1, zn+1) and get the corrected point by averaging slopes yn+1 = yn + h/2 (k1+k2) Complete the following table: The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = Thus the actual value of the function at the point x = 1 is y(1)

Explanation / Answer

n x_n y_n k1 z_n+1 k2 0 0 4 -10 1.5 -3.5 1 0.25 2.3125 -5.53125 0.929688 -1.82422 2 0.5 1.393066 -2.98267 0.6474 -0.8685 3 0.75 0.911671 -1.52918 0.529377 -0.32344 4 1 0.680093 5 y(x) = (2x)/5    - 4/25 y(1) = 0.24

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