Answer
$y(1.5)=8.074$
Work Step by Step
We are given $\frac{dy}{dx}=2x\sqrt 1+y^{2}$
so that $g(x,y)=2x\sqrt 1+y^{2}$
Since $x=1, y=2$
$g(x_{0},y_{0})=2\sqrt 5 \approx 4.472$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=2+4.472\times0.1=2.447$
Now $x_{1}=1.1, y_{1}=2.447$ and $g(x_{1},y_{1})=5.816$
Then $y_{2}=2.447+(5.816)\times0.1=3.029$
$y_{3}=3.029+(7.66)\times0.1=3.795$
$y_{4}=3.795+(10.204)\times0.1=4.815$
$y_{5}=4.815+(13.77)\times0.1=6.192$
$y_{6}=6.192+(18.817)\times0.1=8.074$
