Formula
3. Third order R-K method
`k_1=f(x_0,y_0)`
`k_2=f(x_0+h/2,y_0+(hk_1)/2)`
`k_3=f(x_0+h,y_0+2hk_2-hk_1)`
`y_1=y_0+h/6(k_1+4k_2+k_3)`
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Examples
Find y(0.2) for `y'=(x-y)/2`, `x_0=0, y_0=1`, with step length 0.1 using Runge-Kutta 3 method (1st order derivative)
Solution:
Given `y'=(x-y)/2, y(0)=1, h=0.1, y(0.2)=?`
Third order R-K method
`k_1=f(x_0,y_0)=f(0,1)=-0.5`
`k_2=f(x_0+h/2,y_0+(hk_1)/2)=f(0.05,0.975)=-0.4625`
`k_3=f(x_0+h,y_0+2hk_2-hk_1)=f(0.1,0.9575)=-0.42875`
`y_1=y_0+h/6(k_1+4k_2+k_3)`
`y_1=1+(0.1)/6[-0.5+4(-0.4625)+(-0.42875)]`
`y_1=0.95369`
`:.y(0.1)=0.95369`
Again taking `(x_1,y_1)` in place of `(x_0,y_0)` and repeat the process
`k_1=f(x_1,y_1)=f(0.1,0.95369)=-0.42684`
`k_2=f(x_1+h/2,y_1+(hk_1)/2)=f(0.15,0.93235)=-0.39117`
`k_3=f(x_1+h,y_1+2hk_2-hk_1)=f(0.2,0.91814)=-0.35907`
`y_2=y_1+h/6(k_1+4k_2+k_3)`
`y_2=0.95369+(0.1)/6[-0.42684+4(-0.39117)+(-0.35907)]`
`y_2=0.91451`
`:.y(0.2)=0.91451`
`:.y(0.2)=0.91451`
This material is intended as a summary. Use your textbook for detail explanation.
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