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 The Euler method provides the simplest approach for solving ODEs and most of the more sophisticated techniques derive from it. In the Euler approach, the first order differential equation      can be approximated by making small increments in t and assuming that the slope, f(x,t), is nearly constant over those increments. So, beginning at point x0, the next point x1 becomes      for small dx, which relates to the increment in dt as      then     Repeating for the next point     This can be summarized in the finite difference equation:     The Euler method assumes that the slope f(xn,tn) evaluated at the point (xn,tn) remains approximately constant over the dt interval to x(n+1). See the demonstration of the Euler Method for the case of an object falling in a constant gravitational field. Latest update: Dec.12.2003
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