Homework #$9$: Euler's Method

Consider the following differential equation;

$,y^{\text{'}}=c**y$

$c=1+52$

and

$q,y($x$=0)=1)($

$[20$ points$]$ Obtain the analytical solution via hand calculation

$[30$ points$]$ Determine the value of $y$ in the domain xin$[0,3]$ by using Euler's method with hand calculations. Take the step size $h=0\mathrm{.}5$

$[30$ points$]$ Write a python code for this question with the following goals:

a$.$ the code should give the values of $y$ in the domain using the analytical solution

b$.$ the code should give the values of $y$ in the domain using Euler's method with three different step sizes $h=[0\mathrm{.}5,0\mathrm{.}25,0\mathrm{.}1]$

c$.$ Plot your Euler's method results together with the analytical result

$[20$ points$]$ Comment on the plot you obtained in the last step. What happens with the change in the step size? Which one of the results in $3.$b is the most acceptable and why?

Submit your homework report in a text file format, it must contain your hand calculations, your plots, and your comments. Give your code in the appendix section of your homework report.