Differential Equations: Theory, Technique, and Practice

Preface1 What is a Differential Equation?1.1 Introductory Remarks1.2 The Nature of Solutions1.3 Separable Equations1.4 First-Order Linear Equations1.5 Exact Equations1.6 Orthogonal Trajectories and Families of Curves1.7 Homogeneous Equations1.8 Integrating Factors1.9 Reduction of Order1.9.1 Dependent Variable Missing1.9.2 Independent Variable Missing1.10 The Hanging Chain and Pursuit Curves1.10.1 The Hanging Chain1.10.2 Pursuit Curves1.11 Electrical CircuitsAnatomy of an Application: The Design of a Dialysis MachineProblems for Review and Discovery2 Second-Order Linear Equations2.1 Second-Order Linear Equations with Constant Coefficients2.2 The Method of Undetermined Coefficients2.3 The Method of Variation of Parameters2.4 The Use of a Known Solution to Find Another2.5 Vibrations and Oscillations2.5.1 Undamped Simple Harmonic Motion2.5.2 Damped Vibrations2.5.3 Forced Vibrations2.5.4 A Few Remarks About Electricity2.6 Newton's Law of Gravitation and Kepler's Laws2.6.1 Kepler's Second Law2.6.2 Kepler's First Law2.6.3 Kepler's Third Law2.7 Higher Order Linear Equations, Coupled Harmonic OscillatorsHistorical Note: EulerAnatomy of an Application: Bessel Functions and the Vibrating MembraneProblems for Review and Discovery3 Qualitative Properties and Theoretical Aspects3.1 Review of Linear Algebra3.1.1 Vector Spaces3.1.2 The Concept Linear Independence3.1.3 Bases3.1.4 Inner Product Spaces3.1.5 Linear Transformations and Matrices3.1.6 Eigenvalues and Eigenvectors3.2 A Bit of Theory3.3 Picard's Existence and Uniqueness Theorem3.3.1 The Form of a Differential Equation3.3.2 Picard's Iteration Technique3.3.3 Some Illustrative Examples3.3.4 Estimation of the Picard Iterates3.4 Oscillations and the Sturm Separation Theorem3.5 The Sturm Comparison TheoremAnatomy of an Application: The Green's FunctionProblems for Review and Discovery4 Power Series Solutions and Special Functions4.1 Introduction and Review of Power Series4.1.1 Review of Power Series4.2 Series Solutions of First-Order Differential Equations4.3 Second-Order Linear Equations: Ordinary Points4.4 Regular Singular Points4.5 More on Regular Singular Points4.6 Gauss's Hypergeometric EquationHistorical Note: GaussHistorical Note: AbelAnatomy of an Application: Steady-State Temperature in a BallProblems for Review and Discovery5 Fourier Series: Basic Concepts5.1 Fourier Coefficients5.2 Some Remarks about Convergence5.3 Even and Odd Functions: Cosine and Sine Series5.4 Fourier Series on Arbitrary Intervals5.5 Orthogonal FunctionsHistorical Note: RiemannAnatomy of an Application: Introduction to the Fourier TransformProblems for Review and Discovery6 Partial Differential Equations and Boundary Value Problems6.1 Introduction and Historical Remarks6.2 Eigenvalues, Eigenfunctions, and the Vibrating String6.2.1 Boundary Value Problems6.2.2 Derivation of the Wave Equation6.2.3 Solution of the Wave Equation6.3 The Heat Equation6.4 The Dirichlet Problem for a Disc6.4.1 The Poisson Integral6.5 Sturm-Liouville ProblemsHistorical Note: FourierHistorical Note: DirichletAnatomy of an Application: Some Ideas from Quantum MechanicsProblems for Review and Discovery7 Laplace Transforms7.1 Introduction7.2 Applications to Differential Equations7.3 Derivatives and Integrals of Laplace Transforms7.4 Convolutions7.3.1 Abel's Mechanical Problem7.5 The Unit Step and Impulse FunctionsHistorical Note: LaplaceAnatomy of an Application: Flow Initiated by an Impulsively-Started Flat PlateProblems for Review and Discovery8 The Calculus of Variations8.1 Introductory Remarks8.2 Euler's Equation8.3 Isoperimetric Problems and the Like8.3.1 Lagrange Multipliers8.3.2 Integral Side Conditions8.3.3 Finite Side ConditionsHistorical Note: NewtonAnatomy of an Application: Hamilton's Principle and its ImplicationsProblems for Review and Discovery9 Numerical Methods9.1 Introductory Remarks9.2 The Method of Euler9.3 The Error Term9.4 An Improved Euler Method9.5 The Runge-Kutta MethodAnatomy of an Application: A Constant Perturbation Method for Linear, Second-Order EquationsProblems for Review and Discovery10 Systems of First-Order Equations10.1 Introductory Remarks10.2 Linear Systems10.3 Homogeneous Linear Systems with Constant Coefficients10.4 Nonlinear Systems: Volterra's Predator-Prey EquationsAnatomy of an Application: Solution of Systems with Matrices and ExponentialsProblems for Review and Discovery11 The Nonlinear Theory11.1 Some Motivating Examples11.2 Specializing Down11.3 Types of Critical Points: Stability11.4 Critical Points and Stability for Linear Systems11.5 Stability by Liapunov's Direct Method11.6 Simple Critical Points of Nonlinear Systems11.7 Nonlinear Mechanics: Conservative Systems11.8 Periodic Solutions: The Poincare-Bendixson TheoremHistorical Note: PoincareAnatomy of an Application: Mechanical Analysis of a Block on a SpringProblems for Review and Discovery12 Dynamical Systems12.1 Flows12.1.1 Dynamical Systems12.1.2 Stable and Unstable Fixed Points12.1.3 Linear Dynamics in the Plane12.2 Some Ideas from Topology12.2.1 Open and Closed Sets12.2.2 The Idea of Connectedness12.2.3 Closed Curves in the Plane12.3 Planar Autonomous Systems12.3.1 Ingredients of the Proof of Poincare-BendixsonAnatomy of an Application: Lagrange's EquationsProblems for Review and DiscoveryBibliography