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 3 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)`

Third order Runge-Kutta (RK3) 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,y_n+2hk_2-hk_1,z_n+2hl_2-hl_1)`

`l_3=g(x_n+h,y_n+2hk_2-hk_1,z_n+2hl_2-hl_1)`

`y_(n+1)=y_n+h/6(k_1+4k_2+k_3)`

`z_(n+1)=z_n+h/6(l_1+4l_2+l_3)`

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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,y_0+2hk_2-hk_1,z_0+2hl_2-hl_1)`

`=f(0.1,0.06,0.72)`

`=0.72`

`l_3=g(x_0+h,y_0+2hk_2-hk_1,z_0+2hl_2-hl_1)`

`=g(0.1,0.06,0.72)`

`=-3.12`

Now,
`y_1=y_0+h/6(k_1+4k_2+k_3)`

`=0+0.1/6[1+4(0.8)+(0.72)]`

`=0.082`

`z_1=z_0+1/6(l_1+4l_2+l_3)`

`=1+1/6[-4+4(-3.4)+(-3.12)]`

`=0.6547`

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

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for `n=1,x_1=0.1,y_1=0.082,z_1=0.6547`

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

`=f(0.1,0.082,0.6547)`

`=0.6547`

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

`=g(0.1,0.082,0.6547)`

`=-2.9467`

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

`=f(0.15,0.1147,0.5073)`

`=0.5073`

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

`=g(0.15,0.1147,0.5073)`

`=-2.4883`

`k_3=f(x_1+h,y_1+2hk_2-hk_1,z_1+2hl_2-hl_1)`

`=f(0.2,0.118,0.4517)`

`=0.4517`

`l_3=g(x_1+h,y_1+2hk_2-hk_1,z_1+2hl_2-hl_1)`

`=g(0.2,0.118,0.4517)`

`=-2.2787`

Now,
`y_2=y_1+h/6(k_1+4k_2+k_3)`

`=0.082+0.1/6[0.6547+4(0.5073)+(0.4517)]`

`=0.1343`

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

`:.y(0.2)=0.1343`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`k_3`	`l_3`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	0	1	1	-4	0.8	-3.4	0.72	-3.12	0.1	0.082	0.6547
1	0.1	0.082	0.6547	0.6547	-2.9467	0.5073	-2.4883	0.4517	-2.2787	0.2	0.1343

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