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Problem: Milne's simpson predictor corrector method y'=-2x-y, y(0)=1, x=5 [ Calculator, Method and examples ]

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Your problem `->` Milne's simpson predictor corrector method y'=-2x-y, y(0)=1, x=5


`y'=-2x-y`

Milne's simpson predictor formula is
`y_(n+1,p) = y_(n-3) + (4h)/3 (2y'_(n-2) - y'_(n-1)+ 2y'_(n))`

putting `n=3`, we get

`y_(4,p)=y_0 + (4h)/3 (2y'_1 - y'_2 + 2y'_3) ->(2)`

We have given that
`x_0=0,x_1=1.25,x_2=2.5,x_3=3.75`

and using runge kutta 4 method, we get
`y_0=1,y_1=-0.8075,y_2=-3.0945,y_3=-5.5291`

`y'=-2x-y`

`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 get
`y_(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)=-8.0358`

So, the predicted value is `-8.0358`

Now, we will correct it by corrector method to get the final value
`y'_4=-2x-y=-1.9642` (where `x=5,y=-8.0358`)






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