## M427J Syllabus

DIFFERENTIAL EQUATIONS WITH LINEAR ALGEBRA

**Prerequisite and degree relevance:** The prerequisite is one of M408D, M308L, M408L, M308N, M408S or M408M, with a grade of at least C-.

**Course description: **The syllabus below contains 36-38 hours of required material and two 3 hour optional blocks. As the Fall semester contains 42 hours and the Spring 44 hours, one should adjust accordingly. Instructors should note that it is critical that the full length of allotted time be spent on Chapter 3 and little if any time can be devoted to optional topics.

**Text: ***Differential Equations and Their Applications,* by Martin Braun

**Chapter 1. First-order differential equations [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 [11- 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 (optional)

2.8 Series solutions

2.8.1 Singular points, Euler equations

2.8.2 Regular singular points, the method of Frobenius (optional)

* *

**Chapter 3. Systems of differential equations [15 hours]**

3.1 Algebraic properties of solutions of linear systems

3.2 Vector spaces

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; e^{At}

*Supplement*

3.A Matrix multiplication as linear combination of columns

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

3.C Null and Column spaces

3.D Complete solution set of systems (RREF)

* *

**Optional Chapter 4. Qualitative theory of differential equations [optional 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 [6-7 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

**Optional Chapter 6. Sturm-Liouville boundary value problems [optional 3 hours]**

6.1 Introduction

6.2 Inner product spaces

6.3 Orthogonal bases, Hermitian operators

6.4 Sturm-Liouville theory