#### Answer

The actual result: $4$
The approximation result: $4.8$
Difference: $4.8-4=0.8$

#### Work Step by Step

We are given $\frac{dy}{dx}=4x+3$
Using Euler's method $g(x,y)=4x+3$
Since $x=1, y=0$
$g(x_{0},y_{0})=7$ and
$y_{1}=y_{0}+g(x_{0},y_{0})h=0+7\times0.1=0.7$
Now $x_{1}=1.1, y_{1}=0.7$ and
$g(x_{1},y_{1})=7.4$
Then $y_{2}=0.7+(7.4)\times0.1=1.44$
$y_{3}=1.44+(7.8)\times0.1=2.22$
$y_{4}=2.22+(8.2)\times0.1=3.04$
$y_{5}=3.04+(8.6)\times0.1=3.9$
$y_{6}=3.9+(9)\times0.1=4.8$
$y(1.5)=4.8$
$\frac{dy}{dx}=4x+3$
$\int dy=\int (4x+3)dx$
$y=2x^{2} +3x+C$
With $x=1, y=0$
$C=0-2(1)^{2}-3(1)=-5$
$y=2x^{2} +3x-5$
$y(1.5)=2(1.5)^{2}+3(1.5)-5=4$
Difference between the result and approximation: $4.8-4=0.8$