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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.
Linear Ordinary Differential Equations
First Order Linear Equations
The nth Order Linear Equation
The Wronskian
Non-homogeneous Equations
Complex Solutions
Exercises
Projects
Homogeneous Linear Equations with Constant Coefficients
What to do About Multiple Roots
Euler's Equation
Exercises
Non-homogeneous Equations with Constant Coefficients
How to Calculate a Particular Solution
Exercises
Projects
Boundary Value Problems
Boundary Conditions
Green's Function
Practicalities
Exercises
Separation of Variables
Separable Equations
The Autonomous Case
The Non-autonomous Case
Exercises
One-Parameter Groups of Symmetries
Exercises
Newton's Equation
Motion in a Regular Level Set
Critical Points
Small Oscillations
Exercises
Motion in a Central Force Field
Series Solutions of Linear Equations
Solutions at an Ordinary Point
Preliminaries on Power Series
Solution in Power Series at an Ordinary Point
Exercises
Projects
Solutions at a Regular Singular Point
The Method of Frobenius
The Second Solution When γ-γ Is an Integer
Summary of the Second Solution
The Point at Infinity
Exercises
Projects
Existence Theory
Existence and Uniqueness of Solutions
Picard's Theorem and Successive Approximations
The nth Order Linear Equation Revisited
The First Order Vector Equation
Exercises
Projects
The Exponential of a Matrix
Defining the Exponential
Exercises
Calculation of Matrix Exponentials
Eigenvector Method
Cayley-Hamilton
Interpolation Polynomials
Newton's Divided Differences
Analytic Functions of a Matrix
Exercises
Projects
Linear Systems with Variable Coefficients
Exercises
Projects
Continuation of Solutions
The Maximal Solution
Exercises
Dependence on Initial Conditions
Differentiability of ϕxx
Higher Derivatives of ϕxx
Equations with Parameters
Exercises
Essential Stability Theory
Stability of Equilibrium Points
Lyapunov Functions
Construction of a Lyapunov Function for the Equation dx/dt=Ax
Exercises
Projects
Sturm-Liouville Theory
Symmetry and Self-adjointness
Rayleigh Quotient
Exercises
Eigenvalues and Eigenfunctions
Eigenfunction Expansions
Mean Square Convergence of Eigenfunction Expansions
Eigenvalue Problems with Weights
Exercises
Projects
Afterword