Instead of a final exam, this class has a final project. You will write a short paper about an application of differential equations to a topic of your choice.

Timeline

• Finalize topic: Tuesday, April 27

• Deadline to send draft for comments: Sunday, May 2

• Paper due: Wednesday, May 12

Summary

Differential equations see application in many areas, from ecology to engineering. For the final project for this class, I want you to pick an application and write a 2 to 5 page paper about this application. You should address the phenomenon you are looking at, what assumptions go into deriving your differential equation(s), how you find the general solution to the equation(s), how to find the particular solution given initial value information, and what these solutions tell you about your problem.

For writing your paper, you should consider your classmates as your target audience. You should give enough detail that someone who has taken this class can understand your application and follow the mathematics you present.

You are not required to use this layout, but here is a suggested way to organize your paper into sections.

• Section 1: Introduction Introduce your topic. Explain the phenomenon you are looking at and why it is interesting.

• Section 2: Setting up a differential equation Explain how to set up a differential equation—or possibly multiple equations—which describes your phenomenon. What assumptions underlie the equation?

• Section 3: Solving the equation Give the mathematical work to solve your equation. Explain what is needed to find particular solutions.

• Section 4: Interpreting the solution Now that you’ve solved the differential equation, how do you apply this to say something about your phenomenon? What are the limits of the application?

Topics

You should pick a topic approachable with the methods from this class. (For example, there are many interesting applications of partial differential equations, but those are beyond the scope of this course.) It’s okay if you pick the same topic as someone else in the class, but your work should be your own.

By 11pm on Tuesday, April 27, submit via Gradescope one to two paragraphs explaining your topic, why you picked it, and what resources you plan to use.

Here’s some resources for finding a topic:

• The textbook has examples of applications of differential equations, in chapters 3 and 6. (But don’t pick a topic we covered in class! So no orthogonal trajectories, dilution problems, Newton’s law of cooling, nor harmonic motion.) If you pick an application covered by the textbook, you should pick something from the exercises, i.e. where the work is not all already done for you. Here are a few specific places to look:

  • Lesson 13 has applications to geometry.

  • Lesson 16 has applications to kinematics.

  • Lesson 30 has applications to electric circtuits.

• The library on campus has many books and articles about applications of ODEs to different disciplines. Keywords to search are “[discipline] differential equations” or “[discipline] mathematical modeling”.

Rubric

Here is the breakdown for how your project will be graded:

• Topic Submission: Did you submit your topic choice on Gradescope?

• Format: Does your paper meet the formatting requirements?

• Mechanics: Does your paper adhere to the standards of academic writing? (Grammar, spelling, etc.)

• Differential Equations Content Do you clearly and correctly explain the differential equations content for your project?

• Non-Differential Equations Content Do you adequately explain the context to your project?

Format

Your paper should adhere to the following format guidelines.

• Length: 2–5 pages. Figures and pictures are encouraged where they help explain your ideas, but don’t count for meeting the minimum page length.

• Use a 10pt font, 1.5 inch margins, and single space.

• I won’t enforce a specific citation style, but you should pick a format and use it consistently. If you use LaTeX to write your paper, I strongly encourage you to use bibTeX for managing your citations. If you are unsure which style you want to use, here is a page about Chicago style citations, and you can google to find tools to automatically generate citations.