Teaching Differential Equations: Projects, Resources, and More

1. Teaching Differential Equations and Their Applications Using Projects, Online Resources, and More W. Y. Chan Texas A&M University – Texarkana JMM Baltimore, MD January 19, 2019

2. Outline of the Presentation • Topics Covered in a Differential Equations Course • Projects • Online Resources • More

3. Part 1: Topics Covered in Differential Equations Course

4. Differential equations is one of the required courses for some STEM majors. This course covers techniques for solving differential equations and their applications on various disciplines. To keep the balance of reaching both, homework problems and projects are assigned to help students understand principles and methods.

5. What Majors Need Differential Equations? • Southeast Missouri State University • Computer Science, Mathematics (optional), Physics, and Engineering Physics • Gonzaga University • Computer Science, Mathematics (optional), Physics, Engineering Program (Mechanical, Civil, Computer, Electrical) • Texas A&M University - Texarkana • Computer Science, Mathematics (required), Electrical Engineering • Pre-requisite of Differential Equations is Calculus II • Students taking this course are usually sophomores or juniors.

6. What Majors Need Differential Equations? • Textbook used • W. Boyce and R. DiPrima, Elementary Differential Equations and Boundary Value Problems, Wiley, 8th to 11th editions, 2003-2013. • K. Nagle, E. Saff, and A. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 6th edition, 2011. • R. L. Borrelli and C. S. Coleman, Differential Equations: A Modeling Perspective, Wiley, 2004.

7. Process of Mathematical Modeling

8. Topics Covered in the Course of Differential Equations • First order differential equations, direction fields, principle of the existence of the solution • Different techniques of solving homogeneous first and second differential equations and finding particular solutions • Numerical methods • System of first order differential equations • Laplace Transform • Series of Solution (optional)

9. Some Applications of Differential Equations Radioactive Decay Spring Vibration Falling Objects

10. Topics Covered in the Course of Differential Equations • Applications of DE through introducing mathematical models (radioactive decay, population of entity, LRC electric circuits, oscillating pendulum, spring vibrations, falling object with air resistance) • Modeling with the first order differential equations, compartmental models (mixture problems) , logistic population models • It takes time to discuss the structure of the mathematical models.

11. Falling Objects with Air-Resistance

12. Falling Objects with Air-Resistance

13. The resultant force acting on the object is F1 + F2 = mg + air-resistance • Apply the Newton’s second law

14. Parachute Jumping

15. Parachute Jumping

16. Parachute Jumping Model • Before opening the parachute where k1 = 15N-sec/m. The parachute opens at time 25sec with k2 = 105N-sec/m, The total distance for this fall is 2000m. The values of these ki’s are from textbook or other references.

17. Redesign the Parachute Jumping Model • In 2015, one student was very interested in this mathematical model after it was introduced in class. • This student was a veteran paratrooper. • He served in the US air force at (Fairchild Air Force Base in Washington State) for many years and did the parachute jumping for more than 40 times. • He investigated this problem and created his own mathematical models.

18. Redesign the Parachute Jumping Model - Abstract • USAF at Fairchild Air Force Base are using parachute systems designed for high altitude, high speed egress and openings. These parachutes are designed using parameters that cannot be recreated in a reasonable, safe or fiscally responsible sense. • The objective of his research is to develop differential models for these parachute systems and their functions using known design characteristics, and determine new or confirm actual parameters in the real-world deployment of these parachute systems. He focuses on low-altitude, low-speed parachute deployment under a range of atmospheric conditions.

19. Redesign the Parachute Jumping Model – Mathematical Model • Before opening the parachute The parachute opens at time 4sec,

20. Redesign the Parachute Jumping Model – Mathematical Model • Cm = 1/ft and CD = 1.2/ft are coefficients of drag. • Af is frontal area of parachute: 249.60 ft2 • Am is frontal area of average human: 10.23 ft2 • Vm is volume of average human: 2.34 ft3 • rmis density of average human: 63.68 lb/ft3 • rfis density of air: 7.30 x 10-2 lb/ft3 • The student obtained these parameters through other references or website.

21. His mathematical model was presented in math club meeting (with a real parachute showing to the audiences), MAA sectional meeting, and National Conference on Undergraduate Research in 2015.

22. Part 2: Projects

23. Assigned Projects • Torricelli’s Law of Fluid Flow • Differential Equations and Baseball Batting • Direction Fields and Logistic Models • Optimal Time for Harvesting Catfish • Bungee jumping • Spread of disease using SIR mathematical model

24. Example

25. Example (continued)

26. Example (continued)

27. Solutions and Analysis • Most of the time, students spend time to solve models, analyze them, and understand the behavior of solutions. Then, they make conclusion based on the solutions.

28. Part 3: Online Resources

29. Direction Fields https://www.wolframalpha.com/input/?i=direction+field https://www.geogebra.org/m/W7dAdgqc https://www.geogebra.org/m/QPE4PaDZ https://www.desmos.com/calculator/eijhparfmd • Numerical Solver https://reference.wolfram.com/language/tutorial/NumericalSolutionOfDifferentialEquations.html http://www.math-cs.gordon.edu/~senning/desolver/ • Solving Differential Equations Videos https://www.youtube.com/watch?v=qW1VgnMRNBg https://www.youtube.com/watch?v=z3Ag8WF5M_c https://www.youtube.com/watch?v=pYXpQqvTtEM

30. https://www.geogebra.org/m/W7dAdgqc

31. https://www.desmos.com/calculator/eijhparfmd

32. http://www.math-cs.gordon.edu/~senning/desolver/

33. Part 4: More

34. It would be a challenge to teach the whole process of mathematical modeling.

35. To build a mathematical model to describe a real-world situation, students are also required to • list assumptions, it would take time to write reasonable assumptions; • list all variables; • determine the parameters of the model.

36. Mathematical Modeling Team Project - Example • Heart Rate of Mammals—The following data relate the weights of some mammals to their heart rate in beats per minute. Based on the discussion relating blood flow through the heart to body weight, construct a model relating blood flow through the heart to body weight. Explain the detail to obtain this mathematical model. • This problem and data came from a text.

38. SCUDEM • We joined SCUDEM II and SCUDEM III in 2018. Students are able to practice the mathematical modeling process. SCUDEM II

39. SCUDEM III

40. Some Feedback After SCUDEM After the challenge for SCUDEM, I was able to learn how to create a differential equation using real life event … I learned that there is more ways to make a differential equation than the ones that are just given out of the book, like the homework problems we do for my differential equation class. Nathalie Alvarado, 2018

41. Thank you for your attention.