Answer
The actual result: $3.267$
The approximation: $3.009$
Difference: $3.267-3.009=0.258$
Work Step by Step
We are given $\frac{dy}{dx}=\frac{3}{x}$
Using Euler's method $g(x,y)=\frac{3}{x}$
Since $x=1, y=2$
$g(x_{0},y_{0})=3$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=2+3\times0.1=2.3$
Now $x_{1}=1.1, y_{1}=2.3$ and
$g(x_{1},y_{1})=2.727$
Then $y_{2}=2.3+(2.727)\times0.1=2.572$
$y_{3}=2.572+(2.5)\times0.1=2.822$
$y_{4}=2.822+(2.31)\times0.1=3.053$
$y_{5}=3.053+(2.143)\times0.1=3.267$
$y(1.4)=y_{7}=3.267$
$\frac{dy}{dx}=\frac{3}{x}$
$\int dy=\int \frac{3}{x}dx$
$y=3\ln|x|+C$
With $x=1, y=2$
$C=y-3 \ln1=2$
$y=3\ln|x|+2$
$y(1.4)=3\ln|1.4|+2=3.009$
Difference between the result and approximation: $3.267-3.009=0.258$