Step( i)

$ f(x,y) = \frac{1}{x+y} \,, x_0=0\,,y_0=1\,,h=0.1 \\ x_1=x_0+h=0+0.1=0.1 $

$ k_1 = h f(x_0,y_0)=0.1[f(0,1)]=0.1 $

$ k_2=hf(x_0+\frac{h}{2}, y_0 + \frac{k_1}{2})=0.1f(0.05,1.05)=0.1[\frac{1}{0.05+1.05}]=0.0909 $

$ k_3=hf(x_0+\frac{h}{2}, y_0 + \frac{k_2}{2})=0.1f(0.05,1.0454)=0.1[\frac{1}{0.05+1.0454}]=0.0913 $

$ k_4=hf(x_0+h, y_0 + k_3)=0.1f(0.1,1.0913)=0.1[\frac{1}{0.1+1.0913}]=0.08394 $

$ k=\frac{1}{6}[k_1+2k_2+2k_3+k_4 ] = \frac{1}{6}[ 0.1+ 2(0.0909) +2(0.0913)+0.08394] =0.09139 $

$ y=y_0+k=1+0.9139=1.9139 $

Step (ii)

$x_0=0.1,y_0=1.01939,h=0.1 $

$ k_1=hf(x_0,y_0)=0.1 f(0.1,1.09139)=0.08393 $

$ k_2 =hf(x_0+\frac{h}{2},y_0+\frac{k_1}{2})=0.1f(0.15,1.1333)=0.07792 $

$ k_3 =hf(x_0+\frac{h}{2},y_0+\frac{k_2}{2})=0.1f(0.15,1.13035)=0.07810 $

$ k_4 =hf(x_0+\frac{h}{2},y_0+\frac{k_3}{2})=0.1f(0.2,1.16949)=0.07301 $

$ k=\frac{1}{6}[k_1+2k_2+2k_3+k_4]$

$=\frac{1}{6}[0.08393+2(0.07792)+2(0.07810)+0.07301]=0.1792 $

$ y=y_0+k = 1.09139+0.07816=1.16955$