Answer
$y(0.5)=y_{6}=0.421$
Work Step by Step
We are given $\frac{dy}{dx}=xy+4$
so that $g(x,y)=xy+4$
Since $x=0, y=0$
$g(x_{0},y_{0})=0+4=4$
and $y_{1}=y_{0}+g(x_{0},y_{0})h=0+4\times0.1=0.4$
Now $x_{1}=0.1, y_{1}=0.4$ and $g(x_{1},y_{1})=0.1\times0.4+4=4.04$
Then $y_{2}=0+4.04\times0.1=0.404$
$y_{3}=0+(4.081)\times0.1=0.408$
$y_{4}=0+(4.122)\times0.1=0.412$
$y_{5}=0+(4.165)\times0.1=0.417$
$y_{6}=0+(4.209)\times0.1=0.421$
$y(0.5)$ mean $x=0.5 \rightarrow y_{6}=0.421$
