#### Answer

The actual result: $3.118$
The approximation: $3.139$
Difference : $3.139-3.118=0.021$

#### Work Step by Step

We are given $\frac{dy}{dx}=\frac{2x}{y}$
Using Euler's method $g(x,y)=\frac{2x}{y}$
Since $x=0, y=3$
$g(x_{0},y_{0})=0$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=3+0\times0.1=3$
Now $x_{1}=0.1, y_{1}=3$
and $g(x_{1},y_{1})=0.067$
Then $y_{2}=3+(0.067)\times0.1=3.0067$
$y_{3}=3.0067+(0.133)\times0.1=3.02$
$y_{4}=3.02+(0.199)\times0.1=3.04$
$y_{5}=3.04+(0.263)\times0.1=3.07$
$y_{6}=3.07+(0.39)\times0.1=3.1$
$y_{7}=3.1+(0.387)\times0.1=3.139$
$y(0.4)=3.139$
$\frac{dy}{dx}=\frac{2x}{y}$
$\int ydy=\int xdx$
$\frac{1}{2}y^{2}=x^{2} + C$
With $x=0, y=3$
$C=\frac{9}{2}$
$\frac{1}{2}y^{2}=x^{2} + \frac{9}{2}$
$y(0.6) \rightarrow \frac{1}{2}y^{2}=(0.6)^{2} + \frac{9}{2}$
$\rightarrow y=\frac{9\sqrt 3}{5}\approx 3.118$
Difference between the result and approximation: $3.139-3.118=0.021$