Find y(0.4) for `y''=xz^2-y^2`, `x_0=0, y_0=1, z_0=0`, with step length 0.2 using Runge-Kutta 3 method (second order differential equation)

Solution:
Given `y^('')=xz^2-y^2, y(0)=1, y'(0)=0, h=0.2, y(0.4)=?`

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)=xz^2-y^2=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=1,z_0=0`

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

`=f(0,1,0)`

`=0`

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

`=g(0,1,0)`

`=-1`

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

`=f(0.1,1,-0.1)`

`=-0.1`

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

`=g(0.1,1,-0.1)`

`=-0.999`

`k_3=f(x_0+h,y_0+2hk_2-hk_1,z_0+2hl_2-hl_1)`

`=f(0.2,0.96,-0.1996)`

`=-0.1996`

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

`=g(0.2,0.96,-0.1996)`

`=-0.9136`

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

`=1+0.2/6[0+4(-0.1)+(-0.1996)]`

`=0.98`

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

`=0+1/6[-1+4(-0.999)+(-0.9136)]`

`=-0.197`

`x_1=x_0+h=0+0.2=0.2`

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for `n=1,x_1=0.2,y_1=0.98,z_1=-0.197`

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

`=f(0.2,0.98,-0.197)`

`=-0.197`

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

`=g(0.2,0.98,-0.197)`

`=-0.9527`

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

`=f(0.3,0.9603,-0.2923)`

`=-0.2923`

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

`=g(0.3,0.9603,-0.2923)`

`=-0.8966`

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

`=f(0.4,0.9025,-0.3651)`

`=-0.3651`

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

`=g(0.4,0.9025,-0.3651)`

`=-0.7612`

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

`=0.98+0.2/6[-0.197+4(-0.2923)+(-0.3651)]`

`=0.9223`

`x_2=x_1+h=0.2+0.2=0.4`

`:.y(0.4)=0.9223`

`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	1	0	0	-1	-0.1	-0.999	-0.1996	-0.9136	0.2	0.98	-0.197
1	0.2	0.98	-0.197	-0.197	-0.9527	-0.2923	-0.8966	-0.3651	-0.7612	0.4	0.9223

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