Find y(0.2) for `y''=xz^2-y^2`, `x_0=0, y_0=1, z_0=0`, with step length 0.2 using Runge-Kutta 2 method (2nd order derivative)

Solution:
Given `y^('')=xz^2-y^2, y(0)=1, y'(0)=0, h=0.2, 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)=xz^2-y^2=g(x,y,z)`

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

Second order R-K method for second order differential equation
`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,y_0+hk_1,z_0+hl_1)=f(0.2,1,-0.2)=-0.2`

`l_2=g(x_0+h,y_0+hk_1,z_0+hl_1)=g(0.2,1,-0.2)=-0.992`

`y_1=y_0+(h(k_1+k_2))/2=1-0.02=0.98`

`:.y(0.2)=0.98`

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`:.y(0.2)=0.98`

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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 R-K method for second order differential equation
`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`

`y_1=y_0+hk_2=1-0.02=0.98`

`:.y(0.2)=0.98`

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`:.y(0.2)=0.98`

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