1. MAS222: Differential Equations Semester 1: Ordinary Differential Equations Lecturer: Jonathan Potts (j.potts@Sheffield.ac.uk) Website: http://jonathan-potts.staff.shef.ac.uk/mas222.html

2. Chapter 1. Course information MAS222 is a level 2 course over 2 semesters. The first semester consists of lectures (two per week) and tutorial classes (1 per week). You are required to attend all of these. This is a techniques course, so requires practice. For this, we have provided tutorial sheets and 3 assignment sheets. In the lectures, I will explain when you should be ready to answer these. Note: for those who have already downloaded tutorial sheets 2-9, these may change slightly. I’ve now taken the links down. So please re-download them as they re-appear. Course website: http://jonathan-potts.staff.shef.ac.uk/mas222.html, containing lecture notes, tutorial sheets, assignment sheets, slides etc.http://jonathan-potts.staff.shef.ac.uk/mas222.html Please bring the lecture notes with you to all the lectures. There are empty boxes throughout the notes that need to be filled in either (i) with information written on the board in the lectures, or (ii) given to you as exercises during the lectures. My email: j.potts@sheffield.ac.ukj.potts@sheffield.ac.uk I am away weeks 3 and 4, so Alex Best will be covering my lectures then.

3. Chapter 2. Introduction to Ordinary Differential Equations (ODEs)

4. Introduction to Ordinary Differential Equations (ODEs) A differential equation relates a function and its derivatives Examples:

5. Introduction to Ordinary Differential Equations (ODEs) A differential equation relates a function and its derivatives Examples:

6. Introduction to Ordinary Differential Equations (ODEs) A differential equation relates a function and its derivatives Examples: Here, u is the dependent variable (the variable being differentiated) x and t are independent variables (the variables we are differentiating by)

7. Introduction to Ordinary Differential Equations (ODEs) A differential equation relates a function and its derivatives Examples: Here, u is the dependent variable (the variable being differentiated) x and t are independent variables (the variables we are differentiating by) One independent variable => equation is an ODE (subject of semester 1) More than one => equation is a Partial differential equation (PDE) (semester 2)

8. Derivative as the gradient function of a curve

9. Modelling with ODEs Example 1. A population of organisms.

10. Modelling with ODEs Example 1. A population of organisms. First attempt at a model:

11. Modelling with ODEs Example 1. A population of organisms. First attempt at a model: Second attempt:

12. Modelling with ODEs Example 1. A population of organisms. First attempt at a model: Second attempt: Third attempt:

13. Solving ODEs: initial conditions

14. Initial conditions and chaos

15. Consider the following system: Initial conditions and chaos

16. Classification of ODEs

17. Solving ODEs: Linear, first order, homogeneous

19. Solving ODEs: Linear, first order, inhomogeneous

20. Separable first order ODEs (possibly non- linear)

21. Separable first order ODEs: Example

23. You should now be able to do tutorial sheet 1

24. Chapter 3. Qualitative analysis of ODEs

25. Qualitative analysis of ODEs

26. Direction fields

27. Direction fields: Newton’s law of cooling

28. Direction fields: another example

29. Autonomous equations and phase lines

30. Phase line example

31. You should now be able to do tutorial sheet 2

32. Chapter 4. Planar, first order, autonomous systems of ODEs

33. Definition

34. Example of a planar system

35. Another example of a planar system

36. The simple pendulum Above image from https://en.wikipedia.org/wiki/Phase_portrait Video: https://vimeo.com/53710539

37. Nullclines

38. Example 12

42. You should now be able to do tutorial sheet 3