Formula

Second order Runge-Kutta (RK2) method for second order differential equation formula Method-1 : `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` Method-2 : `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`

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Examples
1. Find y(0.2) for `y''=1+2xy-x^2z`, `x_0=0, y_0=1, z_0=0`, with step length 0.1 using Runge-Kutta 2 method (second order differential equation)

Solution:
Given `y^('')=1+2xy-x^2z, y(0)=1, y'(0)=0, 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)=1+2xy-x^2z=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`

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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)`

`=(0.1)*f(0,1,0)`

`=(0.1)*(0)`

`=0`

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

`=(0.1)*g(0,1,0)`

`=(0.1)*(1)`

`=0.1`

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

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

`=(0.1)*(0.1)`

`=0.01`

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

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

`=(0.1)*(1.199)`

`=0.1199`

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

`=1+0.005`

`=1.005`

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

`=0+0.11`

`=0.11`

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

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for `n=1,x_1=0.1,y_1=1.005,z_1=0.11`

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

`=(0.1)*f(0.1,1.005,0.11)`

`=(0.1)*(0.11)`

`=0.011`

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

`=(0.1)*g(0.1,1.005,0.11)`

`=(0.1)*(1.1999)`

`=0.12`

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

`=(0.1)*f(0.2,1.016,0.2299)`

`=(0.1)*(0.2299)`

`=0.023`

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

`=(0.1)*g(0.2,1.016,0.2299)`

`=(0.1)*(1.3972)`

`=0.1397`

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

`=1.005+0.017`

`=1.022`

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

`:.y(0.2)=1.022`

`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	0.1	0.01	0.1199	0.1	1.005	0.11
1	0.1	1.005	0.11	0.011	0.12	0.023	0.1397	0.2	1.022

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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`

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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)`

`=(0.1)*f(0,1,0)`

`=(0.1)*(0)`

`=0`

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

`=(0.1)*g(0,1,0)`

`=(0.1)*(1)`

`=0.1`

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

`=(0.1)*f(0.05,1,0.05)`

`=(0.1)*(0.05)`

`=0.005`

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

`=(0.1)*g(0.05,1,0.05)`

`=(0.1)*(1.0999)`

`=0.11`

`y_1=y_0+k_2`

`=1+0.005`

`=1.005`

`z_1=z_0+l_2`

`=0+0.11`

`=0.11`

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

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for `n=1,x_1=0.1,y_1=1.005,z_1=0.11`

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

`=(0.1)*f(0.1,1.005,0.11)`

`=(0.1)*(0.11)`

`=0.011`

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

`=(0.1)*g(0.1,1.005,0.11)`

`=(0.1)*(1.1999)`

`=0.12`

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

`=(0.1)*f(0.15,1.0105,0.17)`

`=(0.1)*(0.17)`

`=0.017`

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

`=(0.1)*g(0.15,1.0105,0.17)`

`=(0.1)*(1.2993)`

`=0.1299`

`y_2=y_1+k_2`

`=1.005+0.017`

`=1.022`

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

`:.y(0.2)=1.022`

`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	0.1	0.005	0.11	0.1	1.005	0.11
1	0.1	1.005	0.11	0.011	0.12	0.017	0.1299	0.2	1.022

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