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 4 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)`

Fourth order Runge-Kutta (RK4) method for second order differential equation formula
`k_1=hf(x_n,y_n,z_n)`

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

`k_2=hf(x_n+h/2,y_n+k_1/2,z_n+l_1/2)`

`l_2=hg(x_n+h/2,y_n+k_1/2,z_n+l_1/2)`

`k_3=hf(x_n+h/2,y_n+k_2/2,z_n+l_2/2)`

`l_3=hg(x_n+h/2,y_n+k_2/2,z_n+l_2/2)`

`k_4=hf(x_n+h,y_n+k_3,z_n+l_3)`

`l_4=hg(x_n+h,y_n+k_3,z_n+l_3)`

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

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

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

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

`=(0.2)*f(0,1,0)`

`=(0.2)*(0)`

`=0`

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

`=(0.2)*g(0,1,0)`

`=(0.2)*(-1)`

`=-0.2`

`k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`

`=(0.2)*f(0.1,1,-0.1)`

`=(0.2)*(-0.1)`

`=-0.02`

`l_2=hg(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`

`=(0.2)*g(0.1,1,-0.1)`

`=(0.2)*(-0.999)`

`=-0.1998`

`k_3=hf(x_0+h/2,y_0+k_2/2,z_0+l_2/2)`

`=(0.2)*f(0.1,0.99,-0.0999)`

`=(0.2)*(-0.0999)`

`=-0.02`

`l_3=hg(x_0+h/2,y_0+k_2/2,z_0+l_2/2)`

`=(0.2)*g(0.1,0.99,-0.0999)`

`=(0.2)*(-0.9791)`

`=-0.1958`

`k_4=hf(x_0+h,y_0+k_3,z_0+l_3)`

`=(0.2)*f(0.2,0.98,-0.1958)`

`=(0.2)*(-0.1958)`

`=-0.0392`

`l_4=hg(x_0+h,y_0+k_3,z_0+l_3)`

`=(0.2)*g(0.2,0.98,-0.1958)`

`=(0.2)*(-0.9528)`

`=-0.1906`

Now,
`y_1=y_0+1/6(k_1+2k_2+2k_3+k_4)`

`=1+1/6[0+2(-0.02)+2(-0.02)+(-0.0392)]`

`=0.9801`

`z_1=z_0+1/6(l_1+2l_2+2l_3+l_4)`

`=0+1/6[-0.2+2(-0.1998)+2(-0.1958)+(-0.1906)]`

`=-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=0.9801,z_1=-0.197`

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

`=(0.2)*f(0.2,0.9801,-0.197)`

`=(0.2)*(-0.197)`

`=-0.0394`

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

`=(0.2)*g(0.2,0.9801,-0.197)`

`=(0.2)*(-0.9529)`

`=-0.1906`

`k_2=hf(x_1+h/2,y_1+k_1/2,z_1+l_1/2)`

`=(0.2)*f(0.3,0.9604,-0.2923)`

`=(0.2)*(-0.2923)`

`=-0.0585`

`l_2=hg(x_1+h/2,y_1+k_1/2,z_1+l_1/2)`

`=(0.2)*g(0.3,0.9604,-0.2923)`

`=(0.2)*(-0.8968)`

`=-0.1794`

`k_3=hf(x_1+h/2,y_1+k_2/2,z_1+l_2/2)`

`=(0.2)*f(0.3,0.9509,-0.2866)`

`=(0.2)*(-0.2866)`

`=-0.0573`

`l_3=hg(x_1+h/2,y_1+k_2/2,z_1+l_2/2)`

`=(0.2)*g(0.3,0.9509,-0.2866)`

`=(0.2)*(-0.8796)`

`=-0.1759`

`k_4=hf(x_1+h,y_1+k_3,z_1+l_3)`

`=(0.2)*f(0.4,0.9228,-0.3729)`

`=(0.2)*(-0.3729)`

`=-0.0746`

`l_4=hg(x_1+h,y_1+k_3,z_1+l_3)`

`=(0.2)*g(0.4,0.9228,-0.3729)`

`=(0.2)*(-0.796)`

`=-0.1592`

Now,
`y_2=y_1+1/6(k_1+2k_2+2k_3+k_4)`

`=0.9801+1/6[-0.0394+2(-0.0585)+2(-0.0573)+(-0.0746)]`

`=0.9226`

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

`:.y(0.4)=0.9226`

`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	1	0	0	-0.2	-0.02	-0.1998	-0.02	-0.1958	-0.0392	-0.1906	0.2	0.9801	-0.197
1	0.2	0.9801	-0.197	-0.0394	-0.1906	-0.0585	-0.1794	-0.0573	-0.1759	-0.0746	-0.1592	0.4	0.9226

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