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 4 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)`

Fourth order Runge-Kutta (RK4) 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)`

`k_3=f(x_n+h/2,y_n+(hk_2)/2,z_n+(hl_2)/2)`

`l_3=g(x_n+h/2,y_n+(hk_2)/2,z_n+(hl_2)/2)`

`k_4=f(x_n+h,y_n+hk_3,z_n+hl_3)`

`l_4=g(x_n+h,y_n+hk_3,z_n+hl_3)`

`y_(n+1)=y_n+h/6(k_1+2k_2+2k_3+k_4)`

`z_(n+1)=z_n+h/6(l_1+2l_2+2l_3+l_4)`

---

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`

`k_3=f(x_0+h/2,y_0+(hk_2)/2,z_0+(hl_2)/2)`

`=f(0.05,0.04,0.83)`

`=0.83`

`l_3=g(x_0+h/2,y_0+(hk_2)/2,z_0+(hl_2)/2)`

`=g(0.05,0.04,0.83)`

`=-3.48`

`k_4=f(x_0+h,y_0+hk_3,z_0+hl_3)`

`=f(0.1,0.083,0.652)`

`=0.652`

`l_4=g(x_0+h,y_0+hk_3,z_0+hl_3)`

`=g(0.1,0.083,0.652)`

`=-2.94`

Now,
`y_1=y_0+h/6(k_1+2k_2+2k_3+k_4)`

`=0+0.1/6[1+2(0.8)+2(0.83)+(0.652)]`

`=0.0819`

`z_1=z_0+h/6(l_1+2l_2+2l_3+l_4)`

`=1+0.1/6[-4+2(-3.4)+2(-3.48)+(-2.94)]`

`=0.655`

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

---

for `n=1,x_1=0.1,y_1=0.0819,z_1=0.655`

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

`=f(0.1,0.0819,0.655)`

`=0.655`

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

`=g(0.1,0.0819,0.655)`

`=-2.9475`

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

`=f(0.15,0.1146,0.5076)`

`=0.5076`

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

`=g(0.15,0.1146,0.5076)`

`=-2.489`

`k_3=f(x_1+h/2,y_1+(hk_2)/2,z_1+(hl_2)/2)`

`=f(0.15,0.1072,0.5306)`

`=0.5306`

`l_3=g(x_1+h/2,y_1+(hk_2)/2,z_1+(hl_2)/2)`

`=g(0.15,0.1072,0.5306)`

`=-2.5512`

`k_4=f(x_1+h,y_1+hk_3,z_1+hl_3)`

`=f(0.2,0.1349,0.3999)`

`=0.3999`

`l_4=g(x_1+h,y_1+hk_3,z_1+hl_3)`

`=g(0.2,0.1349,0.3999)`

`=-2.1392`

Now,
`y_2=y_1+h/6(k_1+2k_2+2k_3+k_4)`

`=0.0819+0.1/6[0.655+2(0.5076)+2(0.5306)+(0.3999)]`

`=0.1341`

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

`:.y(0.2)=0.1341`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`k_3`	`l_3`	`k_4`	`l_4`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	0	1	1	-4	0.8	-3.4	0.83	-3.48	0.652	-2.94	0.1	0.0819	0.655
1	0.1	0.0819	0.655	0.655	-2.9475	0.5076	-2.489	0.5306	-2.5512	0.3999	-2.1392	0.2	0.1341

---

---