# 641 3929 1791664 3929 t n y n y n t n12 y n12 y n12 t

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1791.641 3.929 0 1791.664 3.929 0 t n y n y' n t n+1/2 y n+1/2 y' n+1/2 t n y exact Euler's method : We have y n+1 = y n + h f(t n , y n) . Here h = step size = (b -a) / n and f(t n , y n ) is the derivative y' n at t n , since the differential equation is y ' = f(t, y). If Euler's method is selected these calculations are carried out in columns A to C. Modified Euler's method : Euler's method steps across each interval using the estimated slope a the beginning of the interval. A better choice of slope would be to use the slope at the midpoint of the interval. This is done in two stages, as shown below. The subscript n+1/2 is used to denote a quantity associated with a midpoint value. Thus we have y n+1/2 = y n + ½ h f(t n , y n ) (Estimate of y n+1/2 using Euler's method with step = h/2. Column E) y n+1 = y n + h f( t n+1/2 , y n+1/2 ) (like Euler's method except uses y' at midpoint (Column F). Note that y' values are always computed using the t and y values in the two preceding columns.
Calculation Page 3 1791.688 3.929 0 1791.711 3.929 0 1791.734 3.929 0 1791.758 3.929 0