## Math 2070 Syllabus

Text: *Ordinary Differential Equations*, by William A. Adkins and Mark G. Davidson, Springer, 2012, plus a supplement on Fourier Series Methods and student solutions manual for the supplement. The student solutions manual for the text is available from the Springer website for the text.

### Chapter 1: First Order Differential Equations

- 1.1 An Introduction to Differential Equations
- 1.2 Direction Fields
- 1.3 Separable Differential Equations
- 1.4 Linear First Order Equations
- 1.5 Substitutions; Homogeneous and Bernoulli Equations
- 1.6 Exact Equations

### Chapter 2: The Laplace Transform

- 2.1 Laplace Transform Method: Introduction
- 2.2 Definitions, Basic Formulas, and Principles
- 2.3 Partial Fractions: A Recursive Method for Linear Terms
- 2.4 Partial Fractions: A Recursive Method for Irreducible Quadratics
- 2.5 Laplace Inversion
- 2.6 The Linear Spaces
*E*: Special Cases_{q} - 2.7 The Linear Spaces
*E*: The General Case_{q} - 2.8 Convolution

### Chapter 3: Second Order Constant Coefficient Linear Differential Equations

- 3.1 Notation, Definitions, and some Basic Results
- 3.2 Linear Independence
- 3.3 Linear Homogeneous Differential Equations
- 3.4 The Method of Undetermined Coefficients
- 3.6 Spring Systems or 3.7 RCL Circuits

### Chapter 4: Linear Constant Coefficient Differential Equations

- 4.1 Notation, Definitions, and Basic Results
- 4.2 Linear Homogeneous Differential Equations
- 4.3 Nonhomogeneous Differential Equations

### Chapter 5: Second Order Linear Differential Equations

- 5.1 The Existence and Uniqueness Theorem
- 5.2 The Homogeneous Case
- 5.3 The Cauchy-Euler Equations
- 5.5 Reduction of Order
- 5.6 Variation of Parameters

### Chapter 6: Discontinuous Functions and the Laplace Transform

- 6.1 Calculus of Discontinuous Functions
- 6.2 The Heaviside Class
- 6.3 Laplace Transform Method for function in the Heaviside Class
- 6.4 The Dirac Delta Function
- 6.5 Convolution

### Chapter 8: Matrices

- 8.1 Matrix Operations
- 8.2 Systems of Linear Equations
- 8.3 Invertible Matrices
- 8.4 Determinants
- 8.5 Eigenvectors and Eigenvalues

### Chapter 9: Linear Systems of Differential Equations

- 9.1 Introduction
- 9.2 Linear Systems of Differential Equations
- 9.3 The Matrix Exponential Function and its Laplace Transform
- 9.4 Fulmer's Method
- 9.5 Constant Coefficient Linear Systems

### Supplement: Fourier Series Methods

- 10.1 Periodic Functions and Orthogonality Relations
- 10.2 Fourier Series
- 10.3 Convergence of Fourier Series
- 10.4 Fourier Sine Series and Fourier Cosine Series
- 10.5 Operations on Fourier Series
- 10.6 Applications of Fourier Series

William A. Adkins, August 2013. Updated January 2016.