Numerical Methods Course (Ordinary Differential Equations), Assignment

You are strongly encouraged to work collaboratively utilizing github for version control. Coding will be done in report form in the python language using jupyter notebooks. Both code and reporting will be assessed in this course. With jupyter these will be in one .ipynb document. The report should be readable without use of this document. Please do not repeat the numbering used here. While your code need not be as short as possible, you should automate with for loops rather than typing out multiple lines of code when possible. Functions should be short, written to do one thing. Variable names should be descriptive and allow reader to understand what they represent.

1. Eulers Method

1. (a) Using your original Project Part 1 function, f. Graph the solution to the differential equations initial value problem

   dy =f(x)(1y) y(0)=2. dx

   The solution is of the form,

   y(x) = 1 + ce R f(x)dx but you will need to solve for c using the fact that y(0) = 2.

2. (b) Program an algorithm to preform the Euler method. I suggest using an interval for the input and figuring out the outputs at each of those points. Your interval should start at your initial point.

3. (c) Graph the analytic solution and your numerical solution on this interval.

2. Other Numerical Integration

(a) Program an algorithm to preform the Runge-Kutta 4th order Method. (b) Graph the three different solutions on the same interval.

(c) Compute and table the errors at the end of your interval. 3. Start your report and comment on the following.

(a) Describe your findings in words. (b) Discuss why your numerical methods worked or didnt.

(c) Discuss the similarities and differences in the methods.