"This textbook, in its third edition, is a concise introduction to differential equations for students in science and engineering. Emphasis is placed on the methods of solution, on the constructive use of qualitative analysis, and on the interconnections between differential equations and mathematical modeling. It is written in an accessible, easy-to-read format. There are numerous worked-out examples and exercises in each section, along with a wide variety of applications. The text has also been written with affordability and flexibility in mind. The author is an applied mathematician who has written several successful undergraduate and graduate textbooks on mathematical modeling and scientific computing.
This textbook was written as an introductory course in differential equations. While often taught in mathematics departments, it is not uncommon that this topic is also found in engineering departments or other school departments. The expectation is that the students will have taken at least a year of calculus before approaching this subject."
Preface
1 Introduction
1.1 Terminology for Differential Equations
1.2 Solutions and Non-Solutions of Differential Equations .
2 First-Order Equations
2.1 Separable Equations
2.1.1 General Version
2.2 Integrating Factor
2.2.1 General and Particular Solutions
2.2.2 Interesting But Tangentially Useful Topics 1
2.3 Modeling
2.3.1 Mixing
2.3.2 Newton’s Second Law
2.3.3 Logistic Growth or Decay
2.3.4 Newton’s Law of Cooling
2.4 Steady States and Stability
2.4.1 General Version
2.4.2 Sketching the Solution
2.4.3 Parting Comments
3 Second-Order Linear Equations
3.1 Initial Value Problem
3.2 General Solution of a Homogeneous Equation
3.2.1 Linear Independence and the Wronskian
3.3 Solving a Homogeneous Equation
3.3.1 Two Real Roots
3.3.2 One Real Root and Reduction of Order
3.4 Complex Roots
3.4.1 Euler’s Formula and its Consequences
3.4.2 Second Representation
3.4.3 Third Representation
3.5 Summary for Solving a Homogeneous Equation
3.6 Solution of an Inhomogeneous Equation
3.6.1 Non-Uniqueness of a Particular Solution
3.7 The Method of Undetermined Coefficients
3.7.1 Summary Table
3.8 Solving an Inhomogeneous Equation
3.9 Variation of Parameters
3.9.1 The Solution of an IVP
3.10 Linear Oscillator
3.10.1 The Spring Constant
3.10.2 Simple Harmonic Motion
3.10.3 Damping
3.10.4 Forced Motion and Resonance
3.11 Euler Equation
3.11.1 Examples
3.12 Guessing the Title of the Next Chapter
4 Linear Systems
4.1 Linear Systems
4.1.1 Example: Transforming to System Form
4.1.2 General Version
4.2 General Solution of a Homogeneous Equation
4.3 Review of Eigenvalue Problems
4.4 Solving a Homogeneous Equation
4.4.1 Complex-Valued Eigenvalues
4.4.2 Defective Matrix
4.5 Summary for Solving a Homogeneous Equation
4.6 Phase Plane
4.6.1 Examples
4.6.2 Connection with an IVP
4.7 Stability
4.8 Modeling
5 Nonlinear Systems
5.1 Non-Linear Systems
5.1.1 Steady-State Solutions
5.2 Stability
5.2.1 Derivation of the Stability Conditions
5.2.2 Summary
5.2.3 Examples
5.3 Periodic Solutions
5.3.1 Closed Solution Curves and Hamiltonians
5.3.2 Finding the Period
5.4 Motion in a Central Force Field
5.4.1 Steady States
5.4.2 Periodic Orbit
6 Laplace Transform
6.1 Definition
6.1.1 Linearity Property
6.2 Inverse Laplace Transform
6.3 Properties of the Laplace Transform
6.3.1 Transformation of Derivatives
6.3.2 Convolution Theorem
6.4 Solving Differential Equations
6.4.1 The Transfer Function
6.4.2 Comments and Limitations on Using the Laplace Transform
6.5 Jump Discontinuities
6.6 Mathematical Foundations
6.7 Solving Equations with Non-Smooth Forcing
6.7.1 Impulse Forcing
6.8 Solving Linear Systems
6.8.1 Chapter 4 versus Chapter 6
7 Partial Differential Equations
7.1 Balance Laws
7.2 Boundary Value Problems
7.2.1 Eigenvalue Problems
7.3 Separation of Variables
7.3.1 Separation of Variables Assumption
7.3.2 Finding F(x) and λ
7.3.3 Finding G(t)
7.3.4 The General Solution
7.3.5 Satisfying the Initial Condition
7.3.6 Examples
7.4 Sine and Cosine Series
7.4.1 Finding the bn’s
7.4.2 Convergence Theorem
7.4.3 Examples
7.4.4 Cosine Series
7.4.5 Differentiability
7.4.6 Infinite Dimensional
7.5 Wave Equation
7.5.1 Examples
7.5.2 Natural Modes and Standing Waves
7.6 Inhomogeneous Boundary Conditions
7.6.1 Steady State Solution
7.6.2 Transformed Problem
7.6.3 Summary
7.6.4 Wave Equation
7.7 Inhomogeneous PDEs
7.7.1 Summary
7.7.2 A Very Useful Observation
7.8 Laplace’s Equation
7.8.1 Rectangular Domain
7.8.2 Circular Domain
A Matrix Algebra: Summary
B Answers
Bibliography
Index
