The textbook presents a rather unique combination of topics in ODEs, examples and presentation style. The primary intended audience is undergraduate (2nd, 3rd, or 4th year) students in engineering and science (physics, biology, economics). The needed pre-requisite is a mastery of single-variable calculus. A wealth of included topics allows using the textbook in up to three sequential, one-semester ODE courses. Presentation emphasizes the development of practical solution skills by including a very large number of in-text examples and end-of-section exercises. All in-text examples, be they of a mathematical nature or a real-world examples, are fully solved, and the solution logic and flow are explained. Even advanced topics are presented in the same undergraduate-friendly style as the rest of the textbook. Completely optional interactive laboratory-type software is included with the textbook.
Author(s): Victor Henner, Alexander Nepomnyashchy, Tatyana Belozerova, Mikhail Khenner
Publisher: Springer
Year: 2023
Language: English
Pages: 608
City: Cham
Preface
Contents
Chapter 1: Introduction
Chapter 2: First-Order Differential Equations
2.1 Existence and Uniqueness of a Solution
2.2 Integral Curves and Isoclines
2.3 Separable Equations
2.4 Linear First-Order Differential Equations
2.4.1 Homogeneous Linear Equations
2.4.2 Nonhomogeneous Linear Equations: Method of a Parameter Variation
2.4.3 Nonhomogeneous Linear Equations: Method of Integrating Factor
2.4.4 Nonlinear Equations That Can Be Transformed into Linear Equations
2.5 Exact Equations
2.6 Equations Unresolved with Respect to a Derivative
2.6.1 Regular and Irregular Solutions
2.6.2 Lagrange´s Equation
2.6.3 Clairaut´s Equation
2.7 Qualitative Approach for Autonomous First-Order Equations: Equilibrium Solutions and Phase Lines
2.8 Examples of Problems Leading to First-Order Differential Equations
Chapter 3: Differential Equations of Order n > 1
3.1 General Considerations
3.2 Second-Order Differential Equations
3.3 Reduction of Order
3.4 Linear Second-Order Differential Equations
3.4.1 Homogeneous Equations
3.4.2 Reduction of Order for a Linear Homogeneous Equation
3.4.3 Nonhomogeneous Equations
3.5 Linear Second-Order Equations with Constant Coefficients
3.5.1 Homogeneous Equations
3.5.2 Nonhomogeneous Equations: Method of Undetermined Coefficients
3.6 Linear Second-Order Equations with Periodic Coefficients
3.6.1 Hill Equation
3.6.2 Mathieu Equation
3.7 Linear Equations of Order n > 2
3.8 Linear Equations of Order n > 2 with Constant Coefficients
3.9 Euler Equation
3.10 Applications
3.10.1 Mechanical Oscillations
3.10.2 RLC Circuit
3.10.3 Floating Body Oscillations
Chapter 4: Systems of Differential Equations
4.1 General Considerations
4.2 Systems of First-Order Differential Equations
4.3 Systems of First-Order Linear Differential Equations
4.4 Systems of Linear Homogeneous Differential Equations with Constant Coefficients
4.5 Systems of Linear Nonhomogeneous Differential Equations with Constant Coefficients
4.6 Matrix Approach
4.6.1 Homogeneous Systems of Equations
4.6.1.1 Matrix Equation
4.6.1.2 Series Solution for a Constant Matrix A
4.6.1.3 The Case of a Diagonalizable Constant Matrix A
4.6.1.4 The Case of a Non-diagonalizable Constant Matrix A
4.6.2 Nonhomogeneous Systems of Equations
4.6.2.1 The General Case
4.6.2.2 The Case of a Constant Matrix A
4.7 Applications
4.7.1 Charged Particle in a Magnetic Field
4.7.2 Precession of a Magnetic Moment in a Magnetic Field
4.7.3 Spring-Mass System
4.7.4 Mutual Inductance
Chapter 5: Qualitative Methods and Stability of ODE Solutions
5.1 Phase Plane Approach
5.2 Phase Portraits and Stability of Solutions in the Case of Linear Autonomous Systems
5.2.1 Equilibrium Points
5.2.2 Stability: Basic Definitions
5.2.3 Real and Distinct Eigenvalues
5.2.4 Complex Eigenvalues
5.2.5 Repeated Real Eigenvalues
5.2.6 Summary
5.2.7 Stability Diagram in the Trace-Determinant Plane
5.3 Stability of Solutions in the Case of Nonlinear Systems
5.3.1 Definition of Lyapunov Stability
5.3.2 Stability Analysis of Equilibria in Nonlinear Autonomous Systems
5.3.3 Orbital Stability
5.4 Bifurcations and Nonlinear Oscillations
5.4.1 Systems Depending on Parameters
5.4.2 Bifurcations in the Case of Monotonic Instability
5.4.3 Bifurcations in the Case of Oscillatory Instability
5.4.4 Nonlinear Oscillations
Chapter 6: Power Series Solutions of ODEs
6.1 Convergence of Power Series
6.2 Series Solutions Near an Ordinary Point
6.2.1 First-Order Equations
6.2.2 Second-Order Equations
6.3 Series Solutions Near a Regular Singular Point
Chapter 7: Laplace Transform
7.1 Introduction
7.2 Properties of the Laplace Transform
7.3 Applications of the Laplace Transform for ODEs
Chapter 8: Fourier Series
8.1 Periodic Processes and Periodic Functions
8.2 Fourier Coefficients
8.3 Convergence of Fourier Series
8.4 Fourier Series for Nonperiodic Functions
8.5 Fourier Expansions on Intervals of Arbitrary Length
8.6 Fourier Series in Cosine or in Sine Functions
8.7 Examples
8.8 The Complex Form of the Trigonometric Series
8.9 Fourier Series for Functions of Several Variables
8.10 Generalized Fourier Series
8.11 The Gibbs Phenomenon
8.12 Fourier Transforms
Chapter 9: Boundary-Value Problems for Second-Order ODEs
9.1 The Sturm-Liouville Problem
9.2 Examples of Sturm-Liouville Problems
9.3 Nonhomogeneous BVPs
9.3.1 Solvability Condition
9.3.2 The General Solution of Nonhomogeneous Linear Equation
9.3.3 The Green´s Function
Chapter 10: Special Functions
10.1 Gamma Function
10.2 Bessel Functions
10.2.1 Bessel Equation
10.2.2 Bessel Functions of the First Kind
10.2.3 Properties of Bessel Functions
10.2.4 Bessel Functions of the Second Kind
10.2.5 Bessel Functions of the Third Kind
10.2.6 Modified Bessel Functions
10.2.7 Boundary Value Problems and Fourier-Bessel Series
10.2.8 Spherical Bessel Functions
10.2.9 Airy Functions
10.3 Legendre Functions
10.3.1 Legendre Equation and Legendre Polynomials
10.3.2 Fourier-Legendre Series in Legendre Polynomials
10.3.3 Associate Legendre Functions
10.3.4 Fourier-Legendre Series in Associated Legendre Functions
10.4 Elliptic Integrals and Elliptic Functions
10.5 Hermite Polynomials
Chapter 11: Integral Equations
11.1 Introduction
11.2 Introduction to Fredholm Equations
11.3 Iterative Method for the Solution of Fredholm Integral Equations of the Second Kind
11.4 Volterra Equation
11.5 Solution of Volterra Equation with the Difference Kernel Using the Laplace Transform
11.6 Applications
11.6.1 Falling Object
11.6.2 Population Dynamics
11.6.3 Viscoelasticity
Chapter 12: Calculus of Variations
12.1 Functionals: Introduction
12.2 Main Ideas of the Calculus of Variations
12.2.1 Function Spaces
12.2.2 Variation of a Functional
12.2.3 Extrema of a Functional
12.3 The Euler Equation and the Extremals
12.3.1 The Euler Equation
12.3.2 Special Cases of Integrability of the Euler Equation
12.3.2.1 F Does Not Contain x, F = F(y,y′)
12.3.2.2 F Does Not Contain y, F = F(x,y′)
12.3.2.3 F Does Not Contain y′, F = F(x,y)
12.3.2.4 F Depends Only on y′, F = F(y′)
12.3.3 Conditions for the Minimum of a Functional
12.4 Geometrical and Physical Applications
12.4.1 The Brachistochrone Problem
12.4.2 The Tautochrone Problem
12.4.3 The Fermat Principle
12.4.4 The Least Surface Area of a Solid of Revolution
12.4.5 The Shortest Distance Between Two Points on a Sphere
12.5 Functionals That Depend on Several Functions
12.5.1 Euler Equations
12.5.2 Application: The Principle of the Least Action
12.6 Functionals Containing Higher-Order Derivatives
12.7 Moving Boundaries
12.8 Conditional Extremum: Isoperimetric Problems
12.9 Functionals That Depend on a Multivariable Function
12.10 Introduction to the Direct Methods for Variational Problems
12.10.1 Ritz´s Method
12.10.2 Ritz´s Method for Quantum Systems
Chapter 13: Partial Differential Equations
13.1 Introduction
13.2 The Heat Equation
13.2.1 Physical Problems Described by the Heat Equation
13.2.2 The Method of Separation of Variables for One-Dimensional Heat Equation
13.2.3 Heat Conduction Within a Circular Domain
13.3 The Wave Equation
13.3.1 Physical Problems Described by the Wave Equation
13.3.2 Separation of Variables for One-Dimensional Equation
13.3.3 Transverse Oscillations of a Circular Membrane
13.4 The Laplace Equation
13.4.1 Physical Problems Described by the Laplace Equation
13.4.2 BVP for the Laplace Equation in a Rectangular Domain
13.4.3 The Laplace Equation in Polar Coordinates
13.4.4 The Laplace Equation in a Sphere
13.5 Three-Dimensional Helmholtz Equation and Spherical Functions
Chapter 14: Introduction to Numerical Methods for ODEs
14.1 Two Numerical Methods for IVP
14.2 A Finite-Difference Method for Second-Order BVP
14.3 Applications
Appendix A: Existence and Uniqueness of a Solution of IVP for First-Order ODE
Appendix B: How to Use the Accompanying Software
B.1 Examples Using the Program ODE First Order
B.2 Examples Using the Program ODE Second Order
B.3 Examples Using the Program Heat
B.4 Examples Using the Program Waves
B.5 Examples Using the Program FourierSeries
Appendix C: Suggested Syllabi
Suggested Syllabus for a One-Semester Course
Suggested Syllabus for Semester Two
Appendix D: Biographical Notes
Bibliography
Index
