## Math 251

### Ordinary and Partial Differential Equations

**4 credits**

**Blue Book Description:** First- and second-order equations; special functions; Laplace transform solutions; higher order equations; Fourier series; partial differential equations.

**Pre-requisites:** MATH 141

**Pre-requisite for:** MATH 412, MATH 419

**Suggested Textbook:**

Elementary Differential Equations and Boundary Value Problems, 8th Edition, by W. Boyce and R. DiPrima, published by Wiley*Check with your instructor to make sure this is the textbook used for your section.*

**Topics:**

Chapter 1: Introduction

1.1 Some Basic Mathematical Models; Direction fields

1.2 Solutions of Some Differential Equations

1.3 Classification of Differential Equations

Chapter 2: First Order Differential Equations

2.1 Linear Equations; Method of Integrating Factors

2.2 Separable Equations

2.3 Modeling with First Order Equations

2.4 Differences Between Linear and Nonlinear Equations

2.5 Autonomous Equations and Population Dynamics (optional)

2.6 Exact Equations and Integrating Factors

Chapter 3: Second Order Linear Equations

3.1 Homogeneous Equations with Constant Coefficients

3.2 Fundamental Solutions of Linear Homogeneous Equations

3.3 Linear Independence and the Wronskian

3.4 Complex Roots of the Characteristic Equation

3.5 Repeated Roots; Reduction of Order

3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients

3.7 Variation of Parameters (optional)

3.8 Mechanical and Electrical Vibrations

3.9 Forced Vibrations

Chapter 4: Higher Order Linear Equations (optional)

4.1 General Theory of nth Order Linear Equations

4.2 Homogeneous Equations with Constant Coefficients

4.3 The Method of Undetermined Coefficients

4.4 The Method of Variation of Parameters

Chapter 5: Series Solutions of Second Order Linear Equations

5.1 Review of Power Series

5.2 Series Solutions Near an Ordinary Point, Part I

5.3 Series Solutions Near an Ordinary Point, Part II

5.4 Regular Singular Points

5.5 Euler Equations

5.6 Series Solutions Near a Regular Singular Point, Part I (optional)

5.7 Series Solutions Near a Regular Singular Point, Part II (optional)

Chapter 6: The Laplace Transform

6.1 Definition of the Laplace Transform

6.2 Solution of Initial Value Problems

6.3 Step Functions

6.4 Differential Equations with Discontinuous Forcing Functions

6.5 Impulse Functions

6.6 The Convolution Integral (optional)

Chapter 10: Partial Differential Equations and Fourier Series

10.1 Two-Point Boundary Value Problems

10.2 Fourier Series

10.3 The Fourier Convergence Theorem

10.4 Even and Odd Functions

10.5 Separtion of Variables; Heat Conduction in a Rod

10.7 The Wave Equation: Vibrations of an Elastic String