Find y(0.2) for `y''=-4z-4y`, `x_0=0, y_0=0, z_0=1`, with step length 0.1 using Runge-Kutta 2 method (second order differential equation)

Solution:
Given `y^('')=-4z-4y, y(0)=0, y'(0)=1, h=0.1, y(0.2)=?`

put `(dy)/(dx)=z` and differentiate w.r.t. x, we obtain `(d^2y)/(dx^2)=(dz)/(dx)`

We have system of equations
`(dy)/(dx)=z=f(x,y,z)`

`(dz)/(dx)=-4z-4y=g(x,y,z)`

Method-1 : Using formula `k_2=f(x_0+h,y_0+hk_1,z_0+hl_1)`

Second order Runge-Kutta (RK2) method for second order differential equation formula
`k_1=f(x_n,y_n,z_n)`

`l_1=g(x_n,y_n,z_n)`

`k_2=f(x_n+h,y_n+hk_1,z_n+hl_1)`

`l_2=g(x_n+h,y_n+hk_1,z_n+hl_1)`

`y_(n+1)=y_n+(h(k_1+k_2))/2`

`z_(n+1)=z_n+(h(l_1+l_2))/2`

---

for `n=0,x_0=0,y_0=0,z_0=1`

`k_1=f(x_0,y_0,z_0)`

`=f(0,0,1)`

`=1`

`l_1=g(x_0,y_0,z_0)`

`=g(0,0,1)`

`=-4`

`k_2=f(x_0+h,y_0+hk_1,z_0+hl_1)`

`=f(0.1,0.1,0.6)`

`=0.6`

`l_2=g(x_0+h,y_0+hk_1,z_0+hl_1)`

`=g(0.1,0.1,0.6)`

`=-2.8`

`y_1=y_0+(h(k_1+k_2))/2`

`=0+0.08`

`=0.08`

`z_1=z_0+(h(l_1+l_2))/2`

`=1-0.34`

`=0.66`

`x_1=x_0+h=0+0.1=0.1`

---

for `n=1,x_1=0.1,y_1=0.08,z_1=0.66`

`k_1=f(x_1,y_1,z_1)`

`=f(0.1,0.08,0.66)`

`=0.66`

`l_1=g(x_1,y_1,z_1)`

`=g(0.1,0.08,0.66)`

`=-2.96`

`k_2=f(x_1+h,y_1+hk_1,z_1+hl_1)`

`=f(0.2,0.146,0.364)`

`=0.364`

`l_2=g(x_1+h,y_1+hk_1,z_1+hl_1)`

`=g(0.2,0.146,0.364)`

`=-2.04`

`y_2=y_1+(h(k_1+k_2))/2`

`=0.08+0.0512`

`=0.1312`

`x_2=x_1+h=0.1+0.1=0.2`

`:.y(0.2)=0.1312`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	0	1	1	-4	0.6	-2.8	0.1	0.08	0.66
1	0.1	0.08	0.66	0.66	-2.96	0.364	-2.04	0.2	0.1312

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Method-2 : Using formula `k_2=f(x_0+h/2,y_0+(hk_1)/2,z_0+(hl_1)/2)`

Second order Runge-Kutta (RK2) method for second order differential equation formula
`k_1=f(x_n,y_n,z_n)`

`l_1=g(x_n,y_n,z_n)`

`k_2=f(x_n+h/2,y_n+(hk_1)/2,z_n+(hl_1)/2)`

`l_2=g(x_n+h/2,y_n+(hk_1)/2,z_n+(hl_1)/2)`

`y_(n+1)=y_n+hk_2`

`z_(n+1)=z_n+hl_2`

---

for `n=0,x_0=0,y_0=0,z_0=1`

`k_1=f(x_0,y_0,z_0)`

`=f(0,0,1)`

`=1`

`l_1=g(x_0,y_0,z_0)`

`=g(0,0,1)`

`=-4`

`k_2=f(x_0+h/2,y_0+(hk_1)/2,z_0+(hl_1)/2)`

`=f(0.05,0.05,0.8)`

`=0.8`

`l_2=g(x_0+h/2,y_0+(hk_1)/2,z_0+(hl_1)/2)`

`=g(0.05,0.05,0.8)`

`=-3.4`

`y_1=y_0+hk_2`

`=0+0.08`

`=0.08`

`z_1=z_0+hl_2`

`=1-0.34`

`=0.66`

`x_1=x_0+h=0+0.1=0.1`

---

for `n=1,x_1=0.1,y_1=0.08,z_1=0.66`

`k_1=f(x_1,y_1,z_1)`

`=f(0.1,0.08,0.66)`

`=0.66`

`l_1=g(x_1,y_1,z_1)`

`=g(0.1,0.08,0.66)`

`=-2.96`

`k_2=f(x_1+h/2,y_1+(hk_1)/2,z_1+(hl_1)/2)`

`=f(0.15,0.113,0.512)`

`=0.512`

`l_2=g(x_1+h/2,y_1+(hk_1)/2,z_1+(hl_1)/2)`

`=g(0.15,0.113,0.512)`

`=-2.5`

`y_2=y_1+hk_2`

`=0.08+0.0512`

`=0.1312`

`x_2=x_1+h=0.1+0.1=0.2`

`:.y(0.2)=0.1312`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	0	1	1	-4	0.8	-3.4	0.1	0.08	0.66
1	0.1	0.08	0.66	0.66	-2.96	0.512	-2.5	0.2	0.1312

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