Differential Equations II

Time and Location

MWF Period 2 – 113 Little Hall

Office hours

To be arranged

Text

Nagle | Saff | Snider – Fundamentals of Differential Equations and Boundary Value Problems, Edition 9 (or 8 or 7)

Topics and Policies

This course builds upon the foundation provided by its prequel MAP2302: `Elementary Differential Equations’. It will cover chapters 8, 10 and 9 of the adopted text, which we shall study in that indicated order. The three editions listed differ little in the material covered by this course.

In Chapter 8 we seek solutions to linear second-order differential equations, first in the form of power series (which you may already have seen) and then in modified form by a procedure attributed to Frobenius. Many differential equations of importance in mathematics and its applications may be solved in this way; in particular, we shall spend some time with Bessel’s equation, which arises naturally in the solution of numerous physical problems.

In Chapter 10 we are introduced to PDEs (partial dfferential equations) which involve the partial derivatives of functions of two or more variables. Our method for solving such equations will involve symmetry arguments, boundary value problems and Fourier series. The specific PDEs on which we shall spend most of our time – the Laplace equation, the heat equation and the wave equation – feature in applications throughout the physical sciences. Among other things, we shall see that their solution in the presence of certain symmetries often involves Bessel’s equation.

In Chapter 9 we consider systems of dfferential equations (usually linear systems): in theory, these may involve n equations in n unknowns; in practice, n will be 2 or 3 and familiarity with 2×2 and 3×3 matrices and their determinants will be assumed. Included will be a discussion of matrix exponentials, along with methods for calculating them.

Homework will be assigned and discussed, but will not go towards grades. Grades will be assigned on the basis of performance in four in-class tests during the semester. The grading scale will be the `standard’ 90% for A, 80% for B, 70% for C, and so on, with +4% for + grades and -3% for – grades; for example, 77% is the B- threshold and 84% is the B+ threshold.

For various matters of policy, please see ‘Policies plus’ at the Files page.