0345 1 0035 1 00355 1 t n y n y n t n12 y n12 y n12 t

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0.0345100.035100.035510tnyny'ntn+1/2yn+1/2y'n+1/2tnyexactEuler's method: We have yn+1= y n+ h f(tn, yn). Here h = step size = (b -a) / n and f(tn, yn) is thderivative y'n at tn, since the differential equation is y ' = f(t, y). If Euler's method is selected thesecalculations are carried out in columns A to C.Modified Euler's method: Euler's method steps across each interval using the estimated slope atthe 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+1/2is used to denote a quantity associated with a midpoint value. Thus we have yn+1/2= yn+ ½ h f(tn, yn)(Estimate of yn+1/2using Euler's method with step = h/2. Column E)yn+1 = yn+ h f( tn+1/2, yn+1/2)(like Euler's method except uses y' at midpoint (Column F). Note thay' values are always computed using the t and y values in the two preceding columns.
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