This book surveys the broad landscape of differential equations, including elements of partial differential equations (PDEs), and concisely presents the topics of most use to engineers. It introduces each topic with a motivating application drawn from electrical, mechanical, and aerospace engineering. The text has reviews of foundations, step-by-step explanations, and sets of solved problems. It fosters students’ abilities in the art of approximation and self-checking. The book addresses PDEs with and without boundary conditions, which demonstrates strong similarities with ordinary differential equations and clear illustrations of the nature of solutions. Furthermore, each chapter includes word problems and challenge problems. Several extended computing projects run throughout the text.
Table of Contents:
Chapter 1: Introduction
1.1 Foundations
1.2 Classes of Differential Equations
1.3 A Few Other Things
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 2: First-Order Linear Ordinary Differential Equations
2.1 First Order Linear Homogeneous ODEs
2.2 First Order Linear Nonhomogeneous ODEs
2.3 System Stability
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 3: First-Order Nonlinear Ordinary Differential Equations
3.1 Nonlinear Separable ODEs
3.2 The Possibility of Transforming Nonlinear Equations into Linear Ones
3.3 Successive Approximations for Almost Linear Systems
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Computing Projects: Phase 1
Chapter 4: Existence, Uniqueness, and Qualitative Analysis
4.1 Motivation
4.2 Successive Approximations (General Form)
4.3 Existence and Uniqueness Theorem
4.4 Qualitative Analysis
4.5 Stability Revisited
4.6 Local Solutions
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 5: Second-Order Linear Ordinary Differential Equations
5.1 Second Order Linear Time-Invariant Homogeneous ODEs
5.2 Fundamental Solutions
5.3 Second Order Linear Nonhomogeneous ODEs
5.4 Existence and Uniqueness of Solutions to Second-Order Linear ODEs
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 6: Higher-Order Linear Ordinary Differential Equations
6.1 Example: Satellite Orbit Decay
6.2 Higher Order Homogeneous Linear Equations
6.3 Higher Order Nonhomogeneous Linear Equations
6.4 Existence and Uniqueness
6.5 Stability of Higher Order Systems
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Computing Projects: Phase 2
Chapter 7: Laplace Transforms
7.1 Preliminaries
7.2 Introducing the Laplace Transform
7.3 Laplace Transforms of Some Common Functions
7.4 Laplace Transforms of Derivatives
7.5 Laplace Transforms in Homogeneous ODEs
7.6 Laplace Transforms in Nonhomogeneous ODEs
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 8: Systems of First-Order Ordinary Differential Equations
8.1 Preliminaries
8.2 Existence and Uniqueness
8.3 Numerical Integration Methods
8.4 Review of Matrix Algebra
8.5 Linear Systems in State Space Format
8.6 A Brief Look at N-Dimensional Nonlinear Systems
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
Chapter 9: Partial Differential Equations
9.1 The Heat Equation
9.2 The Wave Equation
9.3 Laplace’s Equation
9.4 The Beam Equation
Chapter Summary
Solved Problems
Problems to Solve
Word Problems
Challenge Problems
