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 2 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)`

Method-1 : Using formula `k_2=hf(x_0+h,y_0+k_1,z_0+l_1)`

Second order Runge-Kutta (RK2) 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,y_n+k_1,z_n+l_1)`

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

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

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

---

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,y_0+k_1,z_0+l_1)`

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

`=(0.2)*(-0.2)`

`=-0.04`

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

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

`=(0.2)*(-0.992)`

`=-0.1984`

`y_1=y_0+(k_1+k_2)/2`

`=1-0.02`

`=0.98`

`z_1=z_0+(l_1+l_2)/2`

`=0-0.1992`

`=-0.1992`

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

---

for `n=1,x_1=0.2,y_1=0.98,z_1=-0.1992`

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

`=(0.2)*f(0.2,0.98,-0.1992)`

`=(0.2)*(-0.1992)`

`=-0.0398`

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

`=(0.2)*g(0.2,0.98,-0.1992)`

`=(0.2)*(-0.9525)`

`=-0.1905`

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

`=(0.2)*f(0.4,0.9402,-0.3897)`

`=(0.2)*(-0.3897)`

`=-0.0779`

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

`=(0.2)*g(0.4,0.9402,-0.3897)`

`=(0.2)*(-0.8232)`

`=-0.1646`

`y_2=y_1+(k_1+k_2)/2`

`=0.98-0.0589`

`=0.9211`

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

`:.y(0.4)=0.9211`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	1	0	0	-0.2	-0.04	-0.1984	0.2	0.98	-0.1992
1	0.2	0.98	-0.1992	-0.0398	-0.1905	-0.0779	-0.1646	0.4	0.9211

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Method-2 : Using formula `k_2=hf(x_0+h/2,y_0+k_1/2,z_0+l_1/2)`

Second order Runge-Kutta (RK2) 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)`

`y_(n+1)=y_n+k_2`

`z_(n+1)=z_n+l_2`

---

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`

`y_1=y_0+k_2`

`=1-0.02`

`=0.98`

`z_1=z_0+l_2`

`=0-0.1998`

`=-0.1998`

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

---

for `n=1,x_1=0.2,y_1=0.98,z_1=-0.1998`

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

`=(0.2)*f(0.2,0.98,-0.1998)`

`=(0.2)*(-0.1998)`

`=-0.04`

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

`=(0.2)*g(0.2,0.98,-0.1998)`

`=(0.2)*(-0.9524)`

`=-0.1905`

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

`=(0.2)*f(0.3,0.96,-0.295)`

`=(0.2)*(-0.295)`

`=-0.059`

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

`=(0.2)*g(0.3,0.96,-0.295)`

`=(0.2)*(-0.8955)`

`=-0.1791`

`y_2=y_1+k_2`

`=0.98-0.059`

`=0.921`

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

`:.y(0.4)=0.921`

`n`	`x_n`	`y_n`	`z_n`	`k_1`	`l_1`	`k_2`	`l_2`	`x_(n+1)`	`y_(n+1)`	`z_(n+1)`
0	0	1	0	0	-0.2	-0.02	-0.1998	0.2	0.98	-0.1998
1	0.2	0.98	-0.1998	-0.04	-0.1905	-0.059	-0.1791	0.4	0.921

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