Computers in Physics $-$ Assignment $11$

Euler$$s Technique

The slope of a function y$($x$)$ is described by the differential equation:

$$

$=4+3$

The function has an initial value of y$(0)=1.$

Write a Python notebook which uses Euler$$s Technique to determine the value of the

function at x $=2.$ Use an initial step size of $$x $=0\mathrm{.}5$

Plot the numerical solution and the analytical solution on the same graph.

Notes

$1.$ Include markdown cells to give a suitable description of your work.

$2.$ The code required for this solution is very similar to the example covered in lecture

$11.$ Identify the things you will need to change in that example.

$3.$ Try working out a few iterations by hand so that you have values to use to test your

programme. Build your programme up stage$-$by$-$stage and be happy that each stage

is working before you move on to the next.

$4.$ Your programme should list the values of x and y for each iteration.

$5.$ The analytical solution is easy to find by separating variables, integrating and

applying the initial condition. The solution is $()=22+3+1$

$6.$ Plot the numerical solution and the analytical solution on the same graph for

$$x $=0\mathrm{.}5.$ Include this graph is the version of your notebook that you submit.

$7.$ Re$-$run your programme with $$x $=0\mathrm{.}05$ and $$x $=0\mathrm{.}005.$ In a markdown cell, include

a table of $$x and the corresponding y$(2)$ values.

$8.$ Reset $$x to $0\mathrm{.}5$ and re$-$run all cells. The final version of your notebook should only

list the x and y values, and show the plot, for this initial step size.

$9.$ Download a copy as a html file or a pdf$.$ Use the format which looks best when you

upload it using Turnitin. Submit your homework on the PH$2102$ canvas page before

the deadline.