1. Formula-1 & Example-1
Formula
2. Second order R-K method
Method-1 :
`k_1=hf(x_0,y_0)`
`k_2=hf(x_0+h,y_0+k_1)`
`y_1=y_0+(k_1+k_2)/2`
Method-2 :
`k_1=hf(x_0,y_0)`
`k_2=hf(x_0+h/2,y_0+k_1/2)`
`y_1=y_0+k_2`
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Examples
1. Find y(0.2) for `y'=(x-y)/2`, `x_0=0, y_0=1`, with step length 0.1 using Runge-Kutta 2 method (1st order derivative)
Solution:
Given `y'=(x-y)/2, y(0)=1, h=0.1, y(0.2)=?`
Method-1 : Using formula `k_2=hf(x_0+h,y_0+k_1)`
Second order R-K method
`k_1=hf(x_0,y_0)=(0.1)f(0,1)=(0.1)*(-0.5)=-0.05`
`k_2=hf(x_0+h,y_0+k_1)=(0.1)f(0.1,0.95)=(0.1)*(-0.425)=-0.0425`
`y_1=y_0+(k_1+k_2)/2=1-0.04625=0.95375`
`:.y(0.1)=0.95375`
Again taking `(x_1,y_1)` in place of `(x_0,y_0)` and repeat the process
`k_1=hf(x_1,y_1)=(0.1)f(0.1,0.95375)=(0.1)*(-0.42688)=-0.04269`
`k_2=hf(x_1+h,y_1+k_1)=(0.1)f(0.2,0.91106)=(0.1)*(-0.35553)=-0.03555`
`y_2=y_1+(k_1+k_2)/2=0.95375-0.03912=0.91463`
`:.y(0.2)=0.91463`
`:.y(0.2)=0.91463`
Method-2 : Using formula `k_2=hf(x_0+h/2,y_0+k_1/2)`
Second order R-K method
`k_1=hf(x_0,y_0)=(0.1)f(0,1)=(0.1)*(-0.5)=-0.05`
`k_2=hf(x_0+h/2,y_0+k_1/2)=(0.1)f(0.05,0.975)=(0.1)*(-0.4625)=-0.04625`
`y_1=y_0+k_2=1-0.04625=0.95375`
`:.y(0.1)=0.95375`
Again taking `(x_1,y_1)` in place of `(x_0,y_0)` and repeat the process
`k_1=hf(x_1,y_1)=(0.1)f(0.1,0.95375)=(0.1)*(-0.42688)=-0.04269`
`k_2=hf(x_1+h/2,y_1+k_1/2)=(0.1)f(0.15,0.93241)=(0.1)*(-0.3912)=-0.03912`
`y_2=y_1+k_2=0.95375-0.03912=0.91463`
`:.y(0.2)=0.91463`
`:.y(0.2)=0.91463`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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