Elementary Differential Equations with Boundary Value Problems

Table of Contents

INTRODUCTION TO DIFFERENTIAL EQUATIONS

1.1 Examples of Differential Equations
1.2 Direction Fields


FIRST ORDER DIFFERENTIAL EQUATIONS

2.1 Introduction
2.2 First Order Linear Differential Equations
2.3 Introduction to Mathematical Models
2.4 Population Dynamics and Radioactive Decay
2.5 First Order Nonlinear Differential Equations
2.6 Separable First Order Equations
2.7 Exact Differential Equations
2.8 The Logistic Population Model
2.9 Applications to Mechanics
2.10 Euler’s Method
2.11 Review Exercises


SECOND AND HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

3.1 Introduction
3.2 The General Solution of Homogeneous Equations
3.3 Constant Coefficient Homogeneous Equations
3.4 Real Repeated Roots; Reduction of Order
3.5 Complex Roots
3.6 Unforced Mechanical Vibrations
3.7 The General Solution of a Linear Nonhomogeneous Equation
3.8 The Method of Undetermined Coefficients
3.9 The Method of Variation of Parameters
3.10 Forced Mechanical Vibrations, Electrical Networks, and Resonance
3.11 Higher Order Linear Homogeneous Differential Equations
3.12 Higher Order Homogeneous Constant Coefficient Differential Equations
3.13 Higher Order Linear Nonhomogeneous Differential Equations
3.14 Review Exercises


FIRST ORDER LINEAR SYSTEMS

4.1 Introduction
4.2 Existence and Uniqueness
4.3 Homogeneous Linear Systems
4.4 Constant Coefficient Homogeneous Systems and the Eigenvalue Problem
4.5 Real Eigenvalues and the Phase Plane
4.6 Complex Eigenvalues
4.7 Repeated Eigenvalues
4.8 Nonhomogeneous Linear Systems
4.9 Numerical Methods for Systems of Differential Equations
4.10 The Exponential Matrix and Diagonalization
4.11 Review Exercises


LAPLACE TRANSFORMS

5.1 Introduction
5.2 Laplace Transform Pairs
5.3 The Method of Partial Fractions
5.4 Laplace Transforms of Periodic Functions and System Transfer Functions
5.5 Solving Systems of Differential Equations
5.6 Convolution
5.7 The Delta Function and Impulse Response


NONLINEAR SYSTEMS

6.1 Introduction
6.2 Equilibrium Solutions and Direction Fields
6.3 Conservative Systems
6.4 Stability
6.5 Linearization and the Local Picture
6.6 Two-Dimensional Linear Systems
6.7 Predator-Prey Population Models


NUMERICAL METHODS

7.1 Euler’s Method, Heun’s Method, the Modified Euler’s Method
7.2 Taylor Series Methods
7.3 Runge-Kutta Methods


SERIES SOLUTION OF DIFFERENTIAL EQUATIONS

8.1 Introduction
8.2 Series Solutions near an Ordinary Point
8.3 The Euler Equation
8.4 Solutions Near a Regular Singular Point and the Method of Frobenius
8.5 The Method of Frobenius Continued; Special Cases and a Summary


SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES

9.1 Heat Flow in a Thin Bar. Separation of Variables
9.2 Series Solutions
9.3 Calculating the Solution
9.4 Fourier Series
9.5 The Wave Equation
9.6 Laplace’s Equation
9.7 Higher-Dimensional Problems; Nonhomogeneous Equations


FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS AND THE METHOD OF CHARACTERISTICS

10.1 The Cauchy Problem
10.2 Existence and Uniqueness
10.3 The Method of Characteristics


LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS

11.1 Existence and Uniqueness
11.2 Two-Point Boundary Value Problems for Linear Systems
11.3 Sturm-Liouville Boundary Value Problems