## MATH 209/309 - Linear Analysis

### Suggested Syllabus

**Course description and prerequisite information: **see UW General Catalog

Math 209 serves as the culmination of the Math 207-8-9 program in linear analysis. It combines analytic tools from Math 207 (e.g., general solutions and initial value problems for differential equations; complex numbers and exponentials) with concepts and methods from Math 208 (e.g., eigenvalues and eigenvectors; linear independence, bases, and orthogonal bases). These tools are applied to the solution and qualitative study of linear systems of ordinary differential equations and of boundary value problems for the classical partial differential equations (heat, wave, Laplace). For the latter, separation of variables is used to generate Fourier series solutions.

This syllabus indicates the topics to be covered; instructors may vary the order and emphasis. In the (shorter) winter and spring quarters, choose the smaller number of lectures where a range is given to allow time for a midterm and a second midterm or several short quizzes.

**1. Linear systems of ODE's** (12-13 lectures)

- §§7.1-7.3: Linear algebra (3 lectures)

Review of eigenvalues and eigenvectors, extension to complex matrices, spaces of vector-valued functions, matrix differential equations - §§7.4-7.8: Solving homogeneous linear systems (6 lectures)
- §7.9: Inhomogeneous equations (2 lectures)

Two or three methods from this section - §9.1: Summary of phase plane portraits for linear systems (1 lecture)
- Optional: §§9.2-9.3: Stability of critical points for autonomous nonlinear systems (0-1 lecture)

**2. Fourier series and boundary value problems** (13-15 lectures)

- Introduction
- §10.1: Two-Point Boundary Value Problems (1 lecture)
- §10.5 through equation (18): Separation of variables; one heat equation boundary value problem (2 lectures)
- §§10.2-10.4: Fourier series (4 lectures)
- Boundary Value Problems
- §§10.5-10.6: The heat equation (2 lectures)
- §10.7 The wave equation (2-3 lectures)

Series solutions; D'Alembert's (non-series) solution an option with 3 lectures

- §10.8: The Laplace equation (2-3 lectures)

Dirichlet boundary conditions; other boundary conditions as time permits