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

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

Method-1 : Using formula `k_2=hf(x_0+h,y_0+k_1)`

Second order Runge-Kutta (RK2) method formula
`k_1=hf(x_n,y_n)`

`k_2=hf(x_n+h,y_n+k_1)`

`y_(n+1)=y_n+(k_1+k_2)/2`

---

for `n=0,x_0=0,y_0=-1`

`k_1=hf(x_0,y_0)`

`=(0.1)f(0,-1)`

`=(0.1)*(1)`

`=0.1`

`k_2=hf(x_0+h,y_0+k_1)`

`=(0.1)f(0.1,-0.9)`

`=(0.1)*(0.7)`

`=0.07`

`y_1=y_0+(k_1+k_2)/2`

`=-1+0.085`

`=-0.915`

`x_1=x_0+h=0+0.1=0.1`

---

for `n=1,x_1=0.1,y_1=-0.915`

`k_1=hf(x_1,y_1)`

`=(0.1)f(0.1,-0.915)`

`=(0.1)*(0.715)`

`=0.0715`

`k_2=hf(x_1+h,y_1+k_1)`

`=(0.1)f(0.2,-0.8435)`

`=(0.1)*(0.4435)`

`=0.0444`

`y_2=y_1+(k_1+k_2)/2`

`=-0.915+0.0579`

`=-0.8571`

`x_2=x_1+h=0.1+0.1=0.2`

---

for `n=2,x_2=0.2,y_2=-0.8571`

`k_1=hf(x_2,y_2)`

`=(0.1)f(0.2,-0.8571)`

`=(0.1)*(0.4571)`

`=0.0457`

`k_2=hf(x_2+h,y_2+k_1)`

`=(0.1)f(0.3,-0.8114)`

`=(0.1)*(0.2114)`

`=0.0211`

`y_3=y_2+(k_1+k_2)/2`

`=-0.8571+0.0334`

`=-0.8237`

`x_3=x_2+h=0.2+0.1=0.3`

---

for `n=3,x_3=0.3,y_3=-0.8237`

`k_1=hf(x_3,y_3)`

`=(0.1)f(0.3,-0.8237)`

`=(0.1)*(0.2237)`

`=0.0224`

`k_2=hf(x_3+h,y_3+k_1)`

`=(0.1)f(0.4,-0.8013)`

`=(0.1)*(0.0013)`

`=0.0001`

`y_4=y_3+(k_1+k_2)/2`

`=-0.8237+0.0112`

`=-0.8124`

`x_4=x_3+h=0.3+0.1=0.4`

---

for `n=4,x_4=0.4,y_4=-0.8124`

`k_1=hf(x_4,y_4)`

`=(0.1)f(0.4,-0.8124)`

`=(0.1)*(0.0124)`

`=0.0012`

`k_2=hf(x_4+h,y_4+k_1)`

`=(0.1)f(0.5,-0.8112)`

`=(0.1)*(-0.1888)`

`=-0.0189`

`y_5=y_4+(k_1+k_2)/2`

`=-0.8124-0.0088`

`=-0.8212`

`x_5=x_4+h=0.4+0.1=0.5`

`:.y(0.5)=-0.8212`

`n`	`x_n`	`y_n`	`k_1`	`k_2`	`x_(n+1)`	`y_(n+1)`
0	0	-1	0.1	0.07	0.1	-0.915
1	0.1	-0.915	0.0715	0.0444	0.2	-0.8571
2	0.2	-0.8571	0.0457	0.0211	0.3	-0.8237
3	0.3	-0.8237	0.0224	0.0001	0.4	-0.8124
4	0.4	-0.8124	0.0012	-0.0189	0.5	-0.8212

---

Method-2 : Using formula `k_2=hf(x_0+h/2,y_0+k_1/2)`

Second order Runge-Kutta (RK2) method formula
`k_1=hf(x_n,y_n)`

`k_2=hf(x_n+h/2,y_n+k_1/2)`

`y_(n+1)=y_n+k_2`

---

for `n=0,x_0=0,y_0=-1`

`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`

`y_1=y_0+k_2`

`=-1+0.085`

`=-0.915`

`x_1=x_0+h=0+0.1=0.1`

---

for `n=1,x_1=0.1,y_1=-0.915`

`k_1=hf(x_1,y_1)`

`=(0.1)f(0.1,-0.915)`

`=(0.1)*(0.715)`

`=0.0715`

`k_2=hf(x_1+h/2,y_1+k_1/2)`

`=(0.1)f(0.15,-0.8792)`

`=(0.1)*(0.5792)`

`=0.0579`

`y_2=y_1+k_2`

`=-0.915+0.0579`

`=-0.8571`

`x_2=x_1+h=0.1+0.1=0.2`

---

for `n=2,x_2=0.2,y_2=-0.8571`

`k_1=hf(x_2,y_2)`

`=(0.1)f(0.2,-0.8571)`

`=(0.1)*(0.4571)`

`=0.0457`

`k_2=hf(x_2+h/2,y_2+k_1/2)`

`=(0.1)f(0.25,-0.8342)`

`=(0.1)*(0.3342)`

`=0.0334`

`y_3=y_2+k_2`

`=-0.8571+0.0334`

`=-0.8237`

`x_3=x_2+h=0.2+0.1=0.3`

---

for `n=3,x_3=0.3,y_3=-0.8237`

`k_1=hf(x_3,y_3)`

`=(0.1)f(0.3,-0.8237)`

`=(0.1)*(0.2237)`

`=0.0224`

`k_2=hf(x_3+h/2,y_3+k_1/2)`

`=(0.1)f(0.35,-0.8125)`

`=(0.1)*(0.1125)`

`=0.0112`

`y_4=y_3+k_2`

`=-0.8237+0.0112`

`=-0.8124`

`x_4=x_3+h=0.3+0.1=0.4`

---

for `n=4,x_4=0.4,y_4=-0.8124`

`k_1=hf(x_4,y_4)`

`=(0.1)f(0.4,-0.8124)`

`=(0.1)*(0.0124)`

`=0.0012`

`k_2=hf(x_4+h/2,y_4+k_1/2)`

`=(0.1)f(0.45,-0.8118)`

`=(0.1)*(-0.0882)`

`=-0.0088`

`y_5=y_4+k_2`

`=-0.8124-0.0088`

`=-0.8212`

`x_5=x_4+h=0.4+0.1=0.5`

`:.y(0.5)=-0.8212`

`n`	`x_n`	`y_n`	`k_1`	`k_2`	`x_(n+1)`	`y_(n+1)`
0	0	-1	0.1	0.085	0.1	-0.915
1	0.1	-0.915	0.0715	0.0579	0.2	-0.8571
2	0.2	-0.8571	0.0457	0.0334	0.3	-0.8237
3	0.3	-0.8237	0.0224	0.0112	0.4	-0.8124
4	0.4	-0.8124	0.0012	-0.0088	0.5	-0.8212

---

---