Ordinary Differential Equations: Analytical Methods and Applications

Description

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. 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