# General information

The theory of differential equations is the most important discipline in the whole of modern mathematics.
- Sophus Lie (1895)

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions (…) Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

## Exercise assistant

Truls Bakkejord Ræder
Office 1202, SBII
trulsbak [at] math [dot] ntnu [dot] no

 Main book: D.W. Jordan & P. Smith: Nonlinear Ordinary Differential Equations: An Introduction to for Scientists and Engineers. Fourth edition, Oxford University Press 2007. Chapter 6 from Birkhoff & Rota: Ordinary Differential Equations, John Wiley & Sons 1989 (for sale at the department office on the 7th floor of SB2) H. Hanche-Olsen: Assorted notes on dynamical systems T. Helvik: Drawing phase diagrams with Maple (in Norwegian)

### Scanned lecture notes (Obs: there will be mistakes...)

 4.4.2013 Hamiltonian sysems and index theory Attracting and invariant domains, Lasalles principle Liapunov methods for unstable equilibrium points Continuous dependence on the data for nonlinear ODEs, Projectile problem, Curves and cuvature in the plane General linear nxn systems, matrix exponential, and stability General linear nxn systems, basis, and fundamental matrix Linear 2x2 systems, fundamental matrix, and stability

### Summaries:

#### Intermediate books

• Birkhoff and Rota: Ordinary Differential Equations. John Wiley & Sons 1989.
• Hirsch, Smale, Devaney: Differential equations, dynamical systems and introduction to chaos. Academic Press 2004.
• Verhulst: Nonlinear Differential Equations and Dynamical Systems. Springer 2006.
• Kelley and Peterson: Differential Equations. Classical and Qualitative. Pearson Education 2004.

#### More advanced books and classics

• Perko: Differential equations and dunamical systems. Springer 2001.
• Arnold: Ordinary Differential Equations. Springer 2006.
• Hirsch and Smale: Differential equations, dynamical systems, and linear algebra. Academic Press 1974.
• Hartman: Ordinary Differential Equations. Society for Industrial Mathematics 2002.
• Coddington and Levinson: Theory of ordinary differential equations. McGraw-Hill 1984.

## Final Curriculum

• Jordan & Smith: Ch. 1 (1.6 excluded), 2, 3 (3.2, 3.3 and 3.5 excluded), 8, 10, 11 (11.4 excluded), 13.1.
• Birkhoff & Rota: Ch. 6 (photocopy, chp. 6.11 Analytic equations excluded)
• The note of Hanche-Olsen (see above, chp. 5 excluded).
• The exercises.

(The notes overlap quite a bit with the book. When there is overlap, you may choose which text to read.)

## Exam

• Time: 30.05.2013, 15:00-19:00.
• Aids: Code D (simple calculator)