Find y(0.5) for `y'=-2x-y`, `x_0=0, y_0=-1`, with step length 0.1 using Runge-Kutta 3 method (1st order derivative)

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

Third 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)*(0.85)=0.085`

`k_3=hf(x_0+h,y_0+2k_2-k_1)=(0.1)f(0.1,-0.93)=(0.1)*(0.73)=0.073`

`y_1=y_0+1/6(k_1+4k_2+k_3)`

`y_1=-1+1/6[0.1+4(0.085)+(0.073)]`

`y_1=-0.9145`

`:.y(0.1)=-0.9145`

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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,-0.9145)=(0.1)*(0.7145)=0.0714`

`k_2=hf(x_1+h/2,y_1+k_1/2)=(0.1)f(0.15,-0.8788)=(0.1)*(0.5788)=0.0579`

`k_3=hf(x_1+h,y_1+2k_2-k_1)=(0.1)f(0.2,-0.8702)=(0.1)*(0.4702)=0.047`

`y_2=y_1+1/6(k_1+4k_2+k_3)`

`y_2=-0.9145+1/6[0.0714+4(0.0579)+(0.047)]`

`y_2=-0.8562`

`:.y(0.2)=-0.8562`

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Again taking `(x_2,y_2)` in place of `(x_1,y_1)` and repeat the process

`k_1=hf(x_2,y_2)=(0.1)f(0.2,-0.8562)=(0.1)*(0.4562)=0.0456`

`k_2=hf(x_2+h/2,y_2+k_1/2)=(0.1)f(0.25,-0.8334)=(0.1)*(0.3334)=0.0333`

`k_3=hf(x_2+h,y_2+2k_2-k_1)=(0.1)f(0.3,-0.8351)=(0.1)*(0.2351)=0.0235`

`y_3=y_2+1/6(k_1+4k_2+k_3)`

`y_3=-0.8562+1/6[0.0456+4(0.0333)+(0.0235)]`

`y_3=-0.8224`

`:.y(0.3)=-0.8224`

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Again taking `(x_3,y_3)` in place of `(x_2,y_2)` and repeat the process

`k_1=hf(x_3,y_3)=(0.1)f(0.3,-0.8224)=(0.1)*(0.2224)=0.0222`

`k_2=hf(x_3+h/2,y_3+k_1/2)=(0.1)f(0.35,-0.8113)=(0.1)*(0.1113)=0.0111`

`k_3=hf(x_3+h,y_3+2k_2-k_1)=(0.1)f(0.4,-0.8224)=(0.1)*(0.0224)=0.0022`

`y_4=y_3+1/6(k_1+4k_2+k_3)`

`y_4=-0.8224+1/6[0.0222+4(0.0111)+(0.0022)]`

`y_4=-0.8109`

`:.y(0.4)=-0.8109`

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Again taking `(x_4,y_4)` in place of `(x_3,y_3)` and repeat the process

`k_1=hf(x_4,y_4)=(0.1)f(0.4,-0.8109)=(0.1)*(0.0109)=0.0011`

`k_2=hf(x_4+h/2,y_4+k_1/2)=(0.1)f(0.45,-0.8104)=(0.1)*(-0.0896)=-0.009`

`k_3=hf(x_4+h,y_4+2k_2-k_1)=(0.1)f(0.5,-0.8299)=(0.1)*(-0.1701)=-0.017`

`y_5=y_4+1/6(k_1+4k_2+k_3)`

`y_5=-0.8109+1/6[0.0011+4(-0.009)+(-0.017)]`

`y_5=-0.8196`

`:.y(0.5)=-0.8196`

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`:.y(0.5)=-0.8196`

`n`	`x_n`	`y_n`	`k_1`	`k_2`	`k_3`	`x_(n+1)`	`y_(n+1)`
0	0	-1	0.1	0.085	0.073	0.1	-0.9145
1	0.1	-0.9145	0.0714	0.0579	0.047	0.2	-0.8562
2	0.2	-0.8562	0.0456	0.0333	0.0235	0.3	-0.8224
3	0.3	-0.8224	0.0222	0.0111	0.0022	0.4	-0.8109
4	0.4	-0.8109	0.0011	-0.009	-0.017	0.5	-0.8196

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