2. Find Solution using Bessel's formula

x	f(x)
2.5	24.145
3.0	22.043
3.5	20.225
4.0	18.644
4.5	17.262
5.0	16.047

x = 3.75

Solution:
The value of table for `x` and `y`

x	2.5	3	3.5	4	4.5	5
y	24.145	22.043	20.225	18.644	17.262	16.047

Bessel's method to find solution

`h=3-2.5=0.5`

Taking `x_0=3.5` then `p=(x-x_0)/h=(x-3.5)/0.5`

The difference table is

`x`	`p=(x-3.5)/0.5`	`y`	`Deltay`	`Delta^2y`	`Delta^3y`	`Delta^4y`	`Delta^5y`
2.5	-2	24.145
			-2.102
3	-1	22.043		0.284
			-1.818		-0.047
3.5	0	20.225		0.237		0.009
			-1.581		-0.038		-0.003
4	1	18.644		0.199		0.006
			-1.382		-0.032
4.5	2	17.262		0.167
			-1.215
5	3	16.047

`x = 3.75`

`p = (x - x_0)/h = (3.75 - 3.5)/0.5 = 0.5`

`y_0=20.225, Delta y_0=-1.581,Delta^2y_(-1)=0.237,Delta^3y_(-1)=-0.038,Delta^4y_(-2)=0.009,Delta^5y_(-2)=-0.003`

Bessel's formula is
`y_p=(y_0+y_1)/2+(p-1/2)*Delta y_0 + (p(p-1))/(2!) * (Delta^2y_(-1)+Delta^2y_(0))/2 + ((p-1/2)p(p-1))/(3!) * Delta^3y_(-1) + ((p+1)p(p-1)(p-2))/(4!) * (Delta^4y_(-2)+Delta^4y_(-1))/2 + ((p-1/2)(p+1)p(p-1)(p-2))/(5!) * Delta^5y_(-2)`

`y_(0.5) = (20.225+18.644)/2 + (0.5-1/2)*(-1.581) + (0.5(0.5-1))/(2)*((0.237+0.199))/2 + ((0.5-1/2)0.5(0.5-1))/(6)*(-0.038) + ((0.5+1)0.5(0.5-1)(0.5-2))/(24)*((0.009+0.006))/2 + ((0.5-1/2)(0.5+1)0.5(0.5-1)(0.5-2))/(120)*(-0.003)`

`y_(0.5)=19.4345+0 -0.02725 +0 +0.0001757813 +0`

`y_(0.5)=19.40743`


Solution of Bessel's interpolation is `y(3.75) = 19.40743`

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