Essential Ordinary Differential Equations

Description

This textbook offers an engaging account of the theory of ordinary differential equations intended for advanced undergraduate students of mathematics. Informed by the author’s extensive teaching experience, the book presents a series of carefully selected topics that, taken together, cover an essential body of knowledge in the field. Each topic is treated rigorously and in depth. The book begins with a thorough treatment of linear differential equations, including general boundary conditions and Green’s functions. The next chapters cover separable equations and other problems solvable by quadratures, series solutions of linear equations and matrix exponentials, culminating in Sturm–Liouville theory, an indispensable tool for partial differential equations and mathematical physics. The theoretical underpinnings of the material, namely, the existence and uniqueness of solutions and dependence on initial values, are treated at length. A noteworthy feature of this book is the inclusion of project sections, which go beyond the main text by introducing important further topics, guiding the student by alternating exercises and explanations. Designed to serve as the basis for a course for upper undergraduate students, the prerequisites for this book are a rigorous grounding in analysis (real and complex), multivariate calculus and linear algebra. Some familiarity with metric spaces is also helpful. The numerous exercises of the text provide ample opportunities for practice, and the aforementioned projects can be used for guided study. Some exercises have hints to help make the book suitable for independent study.fsfsfsscs Preface Contents 1 Linear Ordinary Differential Equations 1.1 First Order Linear Equations 1.2 The nth Order Linear Equation 1.2.1 The Wronskian 1.2.2 Non-homogeneous Equations 1.2.3 Complex Solutions 1.2.4 Exercises 1.2.5 Projects 1.3 Homogeneous Linear Equations with Constant Coefficients 1.3.1 What to do About Multiple Roots 1.3.2 Euler's Equation 1.3.3 Exercises 1.4 Non-homogeneous Equations with Constant Coefficients 1.4.1 How to Calculate a Particular Solution 1.4.2 Exercises 1.4.3 Projects 1.5 Boundary Value Problems 1.5.1 Boundary Conditions 1.5.2 Green's Function Practicalities 1.5.3 Exercises 2 Separation of Variables 2.1 Separable Equations 2.1.1 The Autonomous Case 2.1.2 The Non-autonomous Case 2.1.3 Exercises 2.2 One-Parameter Groups of Symmetries 2.2.1 Exercises 2.3 Newton's Equation 2.3.1 Motion in a Regular Level Set 2.3.2 Critical Points Small Oscillations 2.3.3 Exercises 2.4 Motion in a Central Force Field 3 Series Solutions of Linear Equations 3.1 Solutions at an Ordinary Point 3.1.1 Preliminaries on Power Series 3.1.2 Solution in Power Series at an Ordinary Point 3.1.3 Exercises 3.1.4 Projects 3.2 Solutions at a Regular Singular Point 3.2.1 The Method of Frobenius 3.2.2 The Second Solution When γ1-γ2 Is an Integer Summary of the Second Solution 3.2.3 The Point at Infinity 3.2.4 Exercises 3.2.5 Projects 4 Existence Theory 4.1 Existence and Uniqueness of Solutions 4.1.1 Picard's Theorem and Successive Approximations 4.1.2 The nth Order Linear Equation Revisited 4.1.3 The First Order Vector Equation 4.1.4 Exercises 4.1.5 Projects 5 The Exponential of a Matrix 5.1 Defining the Exponential 5.1.1 Exercises 5.2 Calculation of Matrix Exponentials 5.2.1 Eigenvector Method 5.2.2 Cayley-Hamilton 5.2.3 Interpolation Polynomials 5.2.4 Newton's Divided Differences 5.2.5 Analytic Functions of a Matrix 5.2.6 Exercises 5.2.7 Projects 5.3 Linear Systems with Variable Coefficients 5.3.1 Exercises 5.3.2 Projects 6 Continuation of Solutions 6.1 The Maximal Solution 6.1.1 Exercises 6.2 Dependence on Initial Conditions 6.2.1 Differentiability of ϕx0x 6.2.2 Higher Derivatives of ϕx0x 6.2.3 Equations with Parameters 6.2.4 Exercises 6.3 Essential Stability Theory 6.3.1 Stability of Equilibrium Points 6.3.2 Lyapunov Functions 6.3.3 Construction of a Lyapunov Function for the Equation dx/dt=Ax 6.3.4 Exercises 6.3.5 Projects 7 Sturm-Liouville Theory 7.1 Symmetry and Self-adjointness 7.1.1 Rayleigh Quotient 7.1.2 Exercises 7.2 Eigenvalues and Eigenfunctions 7.2.1 Eigenfunction Expansions 7.2.2 Mean Square Convergence of Eigenfunction Expansions 7.2.3 Eigenvalue Problems with Weights 7.2.4 Exercises 7.2.5 Projects Afterword Index