Step 1

We have `x_0=0` and `y_0=5`.

`F_1=hf(x,y)` ` = 0.2((0+5)sin (0)(5))` ` = 0`

For `F_2`, we need to know:

`x+h/2 = 0+0.2/2 = 0.1`, and

`y+F_1/2 = 5+0/2=5 `

We substitute these into the `F_2` expression:

`F_2=hf(x+h/2,y+F_1/2)` ` = 0.2((0.1+5)sin (0.1)(5))` `=0.48901404937`

For `F_3`, we need to know:

`y+F_2/2` ` = 5+0.48901404937/2` `=5.24450702469`

So

`F_3=hf(x+h/2,y+F_2/2) `

`= 0.2((0.1+5.24450702469)` `{:xxsin (0.1)(5.24450702469))`

`=0.53523913352`

For `F_4`, we need to know:

`y+F_3` ` = 5+0.53523913352` ` = 5.53523913352 `

So

`F_4=hf(x+h,y+F_3)`

` = 0.2((0.2+5.53523913352)` `{:xxsin (0.2)(5.53523913352))`

`= 1.02589900571`

Step 2

Next, we take those 4 values and substitute them into the Runge-Kutta RK4 formula:

`y(x+h)=y(x)` `+1/6(F_1+2F_2+2F_3+F_4)`

`=5+1/6(0+ ` `2xx0.48901404937+ ` `2xx0.53523913352 ` `{:+ 1.02589900571)`

`=5.5124008953`

As before, we need to take this `y_1` value and use the new `x_1=0.2` value to find the next value, `y_2`, and so on up to `x=2`.

Following is the table of resulting values.