1. `y'=(x+y)/2`,

`x_i`	0	0.5	1	1.5
`y_i`	2	2.636	3.595	4.968

Find y(2) by Adams bashforth predictor method

Solution:
`y'=(x+y)/2`

Adam'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 get

`y_(4,p)=y_3 + h/24 (55y'_(3) - 59y'_2 + 37y'_1 - 9y'_0) ->(2)`

We have given that
`x_0=0,x_1=0.5,x_2=1,x_3=1.5`

`y_0=2,y_1=2.636,y_2=3.595,y_3=4.968`

`y'=(x+y)/2`

`y'_0=(x+y)/2=1` (where `x=0,y=2`)

`y'_1=(x+y)/2=1.568` (where `x=0.5,y=2.636`)

`y'_2=(x+y)/2=2.2975` (where `x=1,y=3.595`)

`y'_3=(x+y)/2=3.234` (where `x=1.5,y=4.968`)

putting the values in (2), we get
`y_(4,p)=y_3 + h/24 (55y'_(3) - 59y'_2 + 37y'_1 - 9y'_0)`

`y_(4,p)=4.968 + 0.5/24 * (55 * 3.234 - 59 * 2.2975 + 37 * 1.568 - 9 * 1)`

`y_(4,p)=4.968 + 0.5/24 * (177.87 - 135.5525 + 58.016 - 9)`

`y_(4,p)=4.968 + 0.5/24 * (91.3335)`

`y_(4,p)=4.968 +1.9028`

`y_(4,p)=6.8708`

So, the predicted value is `6.8708`

Now, we will correct it by corrector method to get the final value
`y'_4=(x+y)/2=4.4354` (where `x=2,y=6.8708`)

Adam's Bashforth Corrector formula is
`y_(n+1,c) = y_n + h/24 (9y'_(n+1) + 19y'_(n) - 5y'_(n-1) + y'_(n-2))`

putting `n=3`, we get

`y_(4,c) = y_3 + h/24 (9y'_4 + 19y'_3 - 5y'_2 + y'_1)`

`y_(4,c) = 4.968 + 0.5/24 * (9 * 4.4354 + 19 * 3.234 - 5 * 2.2975 + 1.568)`

`y_(4,c) = 4.968 + 0.5/24 * (39.9185 + 61.446 - 11.4875 + 1.568)`

`y_(4,c) = 4.968 + 0.5/24 * (91.445)`

`y_(4,c) = 4.968 +1.9051`

`y_(4,c)=6.8731`

`:.y(2) = 6.8731`