E-Mail: arbogast@ices.utexas.edu

Office: RLM 11.162, Phone: 512-471-0166

Office hours: MTWTh 8:30-9:30 a.m.

E-mail: rfayvisovich@math.utexas.edu

Office: RLM 11.152, Phone: 512-471-1684

Office hours: T 2:00-4:00 p.m.

Chapter 1. First-order differential equations (supplement with direction fields) [4 hours]

1.1 Introduction

1.2 First-order linear differential equations

1.4 Separable equations

1.9 Exact equations, and why we cannot solve very many differential equations

1.10 The existence-uniqueness theorem; Picard iteration

Chapter 2. Second-order linear differential equations [12 hours]

2.1 Algebraic properties of solutions

2.2 Linear equations with constant coefficients

2.2.1 Complex roots

2.2.2 Equal roots; reduction of order

2.3 The nonhomogeneous equation

2.4 The method of variation of parameters

2.5 The method of judicious guessing

2.6 Mechanical vibrations

2.8 Series solutions

2.8.1 Singular points, Euler equations

2.8.2 Regular singular points, the method of Frobenius

Chapter 3. Systems of differential equations (with supplementary material added) [14 hours]

3.1 Algebraic properties of solutions of linear systems

3.A Matrix multiplication as linear combinations of columns

3.2 Vector spaces

3.B Vectors as arrows in **R ^{n}** and a geometric meaning of operations

3.C Complete solution system of linear equations (RREF)

3.D Null and column spaces of a matrix

3.3 Dimension of a vector space

3.4 Applications of linear algebra to differential equations

3.5 The theory of determinants

3.6 Solutions of simultaneous linear equations

3.7 Linear transformations

3.8 The eigenvalue-eigenvector method of finding solutions

3.9 Complex roots

3.10 Equal roots

3.11 Fundamental matrix solutions;

Chapter 4. Qualitative theory of differential equations [3 hours]

4.1 Introduction

4.2 Stability of linear systems

4.4 The phase-plane

4.7 Phase portraits of linear systems

Chapter 5. Separation of variables and Fourier series [8 hours]

5.1 Two point boundary-value problems

5.2 Introduction to partial differential equations

5.3 The heat equation; separation of variables

5.4 Fourier series

5.5 Even and odd functions

5.6 Return to the heat equation

Chapter 6. Sturm-Liouville boundary value problems [optional 3 hours, if time permits]

6.1 Introduction

6.2 Inner product spaces

6.3 Orthogonal bases, Hermitian operators

6.4 Sturm-Liouville theory