Find y(0.2) for `y'=-y`, `x_0=0, y_0=1`, with step length 0.1 using Runge-Kutta 4 method (1st order derivative)
Solution:
Given `y'=-y, y(0)=1, h=0.1, y(0.2)=?`
Forth order R-K method
`k_1=f(x_0,y_0)=f(0,1)=-1`
`k_2=f(x_0+h/2,y_0+(hk_1)/2)=f(0.05,0.95)=-0.95`
`k_3=f(x_0+h/2,y_0+(hk_2)/2)=f(0.05,0.9525)=-0.9525`
`k_4=f(x_0+h,y_0+hk_3)=f(0.1,0.90475)=-0.90475`
`y_1=y_0+h/6(k_1+2k_2+2k_3+k_4)`
`y_1=1+0.1/6[-1+2(-0.95)+2(-0.9525)+(-0.90475)]`
`y_1=0.90484`
`:.y(0.1)=0.90484`
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(0.1,0.90484)=-0.90484`
`k_2=f(x_1+h/2,y_1+(hk_1)/2)=f(0.15,0.8596)=-0.8596`
`k_3=f(x_1+h/2,y_1+(hk_2)/2)=f(0.15,0.86186)=-0.86186`
`k_4=f(x_1+h,y_1+hk_3)=f(0.2,0.81865)=-0.81865`
`y_2=y_1+h/6(k_1+2k_2+2k_3+k_4)`
`y_2=0.90484+0.1/6[-0.90484+2(-0.8596)+2(-0.86186)+(-0.81865)]`
`y_2=0.81873`
`:.y(0.2)=0.81873`
`:.y(0.2)=0.81873`
This material is intended as a summary. Use your textbook for detail explanation.
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