Answer
Difference between the actual result and approximation: $5.023-4.759=0.264$
Work Step by Step
We are given $\frac{dy}{dx}=2xy$
Using Euler's method $g(x,y)=2xy$
Since $x=1, y=1$
$g(x_{0},y_{0})=2$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=1+2\times0.1=1.2$
Now $x_{1}=1.1, y_{1}=1.2$ and $g(x_{1},y_{1})=2.64$
Then $y_{2}=1.2+(2.64)\times0.1=1.464$
$y_{3}=1.464+(3.5136)\times0.1=1.8153$
$y_{4}=1.8153+(4.72)\times0.1=2.287$
$y_{5}=2.287+(6.4036)\times0.1=2.927$
$y_{6}=2.927+(8.782)\times0.1=3.8052$
$y_{7}=3.8052+(12.177)\times0.1=5.023$
$y(1.6)=y_{7}=5.023$
$\frac{dy}{dx}=2xy$
$\int\frac{1}{y}dy=\int2x dx$
$\ln|y|=x^{2}+C$
With $y(1)=1$
$C=\ln|y|-x^{2}$
$C=-1$
$\rightarrow \ln|y|=x^{2}-1$
$y(1.6) \rightarrow \ln|y|=(1.6)^{2}-1$
$\ln|y|=1.56$
$|y|=e^{1.56}\approx4.759$
Difference between the actual result and approximation: $5.023-4.759=0.264$
