Answer
$y(0.5)$ mean $x=0.5 \rightarrow y_{6}=2+(7.846)\times0.1=2.785$
Work Step by Step
We are given $\frac{dy}{dx}=x^{2}+y^{2}$
so that $g(x,y)=x^{2}+y^{2}$
Since $x=0, y=2$
$g(x,y)=0^{2}+2^{2}=4$
and $y_{1}=y_{0}+g(x_{0},y_{0})h=2+4\times0.1=2.4$
Now $x_{1}=0.1, y_{1}=2.4$ and $g(x_{1},y_{1})=0.1^{2}+2.4^{2}=5.77$
Then $y_{2}=2+5.77\times0.1=2.577$
$y_{3}=2+(6.681)\times0.1=2.668$
$y_{4}=2+(7.209)\times0.1=2.721$
$y_{5}=2+(7.564)\times0.1=2.756$
$y_{6}=2+(7.846)\times0.1=2.785$
$y(0.5)$ mean $x=0.5 \rightarrow y_{6}=2+(7.846)\times0.1=2.785$
