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 Problem: Milne's simpson predictor corrector method y'=-2x-y, y(0)=1, x=5 [ Calculator, Method and examples ]Solution:Your problem -> Milne's simpson predictor corrector method y'=-2x-y, y(0)=1, x=5y'=-2x-yMilne's simpson predictor formula isy_(n+1,p) = y_(n-3) + (4h)/3 (2y'_(n-2) - y'_(n-1)+ 2y'_(n))putting n=3, we gety_(4,p)=y_0 + (4h)/3 (2y'_1 - y'_2 + 2y'_3) ->(2)We have given thatx_0=0,x_1=1.25,x_2=2.5,x_3=3.75and using runge kutta 4 method, we gety_0=1,y_1=-0.8075,y_2=-3.0945,y_3=-5.5291y'=-2x-yy'_1=-2x-y=-1.6925 (where x=1.25,y=-0.8075)y'_2=-2x-y=-1.9055 (where x=2.5,y=-3.0945)y'_3=-2x-y=-1.9709 (where x=3.75,y=-5.5291)putting the values in (2), we gety_(4,p)=y_0 + (4h)/3 (2y'_1 - y'_2 + 2y'_3) ->(2)y_(4,p)=1 + (4*1.25)/3 * (2 * -1.6925 - -1.9055 + 2 * -1.9709)y_(4,p)=-8.0358So, the predicted value is -8.0358Now, we will correct it by corrector method to get the final valuey'_4=-2x-y=-1.9642 (where x=5,y=-8.0358)

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