Content: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.

However, this theory is not only of interest to the applied mathematician: ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology. The first half of this module will focus on ordinary differential equations - how to understand them and how to solve them. The second half of the module covers topics from multivariable calculus - partial derivatives, div, grad, curl, and some differential geometry and integration needed for subsequent modules on differential equations.

Aims: To introduce simple differential equations and methods for their solution and to provide a solid foundation in the calculus needed to study future modules involving ordinary and partial differential equations.

Objectives: You should be able to solve various simple differential equations (first order, linear second order and coupled systems of first order equations), be able to manipulate div, grad, and curl operations, and be able to integrate over simple curves and surfaces.

Books:

The primary text will be:
J. C. Robinson An Introduction to Ordinary Differential Equations, Cambridge University Press 2003.
J. Stewart, D. Clegg, S. Watson, Multivariable Calculus, ninth edition, metric version, Brooks/Cole 2022.

Additional references are:
W. Boyce and R. Di Prima, Elementary Differential Equations and Boundary Value Problems, Wiley 1997.