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 Problem: Adams bashforth predictor method y'=-2x-y, y(0)=1, x=5 [ Calculator, Method and examples ]Solution:Your problem -> Adams bashforth predictor method y'=-2x-y, y(0)=1, x=5y'=-2x-yAdam's Bashforth Predictor formula is y_(n+1,p) = y_n + h/24 (55y'_(n) - 59y'_(n-1) + 37y'_(n-2) - 9y'_(n-3))putting n=3, we gety_(4,p)=y_3 + h/24 (55y'_(3) - 59y'_2 + 37y'_1 - 9y'_0) ->(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'_0=-2x-y=-1 (where x=0,y=1)y'_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_3 + h/24 (55y'_(3) - 59y'_2 + 37y'_1 - 9y'_0)y_(4,p)=-5.5291 + 1.25/24 * (55 * -1.9709 - 59 * -1.9055 + 37 * -1.6925 - 9 * -1)y_(4,p)=-8.1125So, the predicted value is -8.1125Now, we will correct it by corrector method to get the final valuey'_4=-2x-y=-1.8875 (where x=5,y=-8.1125)

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