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Find Runge-Kutta 4 method y'=-2x-y, y(0)=1, xn=0.5, h=0.1

Solution:
Your problem `->` Runge-Kutta 4 method y'=-2x-y, y(0)=1, xn=0.5, h=0.1


Given `y'=-2x-y, y(0)=1, h=0.1, y(0.5)=?`

Forth order R-K method
`k_1=hf(x_0,y_0)=(0.1)f(0,1)=(0.1)*(-1)=-0.1`

`k_2=hf(x_0+h/2,y_0+k_1/2)=(0.1)f(0.05,0.95)=(0.1)*(-1.05)=-0.105`

`k_3=hf(x_0+h/2,y_0+k_2/2)=(0.1)f(0.05,0.9475)=(0.1)*(-1.0475)=-0.1048`

`k_4=hf(x_0+h,y_0+k_3)=(0.1)f(0.1,0.8952)=(0.1)*(-1.0952)=-0.1095`

`y_1=y_0+1/6(k_1+2k_2+2k_3+k_4)`

`y_1=1+1/6[-0.1+2(-0.105)+2(-0.1048)+(-0.1095)]`

`y_1=0.8952`

`:.y(0.1)=0.8952`

Again taking `(x_1,y_1)` in place of `(x_0,y_0)` repeat the process






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