Fundamentals of Differential Equations and Boundary Value Problems – R. Nagle, E. Saff, D. Snider – 5th Edition

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Fundamentals of presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. Available in two versions, these flexible texts offer the instructor many choices in syllabus design, emphasis (theory, methodology, applications, and numerical methods), and in using commercially available software.

Fundamentals of Differential Equations, Seventh Edition is suitable for a one-semester sophomore- or junior-level course. Fundamentals of with Boundary Value Problems, Fifth Edition, contains enough material for a two-semester that covers and builds on boundary value problems. The Boundary Value version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; of Autonomous Systems; and Existence and Uniqueness Theory).

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  • 1. Introduction
    1.1 Background
    1.2 Solutions and Initial Value Problems
    1.3 Direction Fields
    1.4 The Approximation Method of Euler
    2. First-Order Differential Equations
    2.1 Introduction: Motion of a Falling Body
    2.2 Separable Equations
    2.3 Linear Equations
    2.4 Exact Equations
    2.5 Special Integrating Factors
    2.6 Substitutions and Transformations
    3. Mathematical Models and Numerical Methods Involving First Order Equations
    3.1 Mathematical Modeling
    3.2 Compartmental Analysis
    3.3 Heating and Cooling of Buildings
    3.4 Newtonian Mechanics
    3.5 Electrical Circuits
    3.6 Improved Euler's Method
    3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta
    4. Linear Second-Order Equations
    4.1 Introduction: The Mass-Spring Oscillator
    4.2 Homogeneous Linear Equations: The General Solution
    4.3 Auxiliary Equations with Complex Roots
    4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients
    4.5 The Superposition Principle and Undetermined Coefficients Revisited
    4.6 Variation of Parameters
    4.7 Variable-Coefficient Equations
    4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
    4.9 A Closer Look at Free Mechanical Vibrations
    4.10 A Closer Look at Forced Mechanical Vibrations
    5. Introduction to Systems and Phase Plane Analysis
    5.1 Interconnected Fluid Tanks
    5.2 Elimination Method for Systems with Constant Coefficients
    5.3 Solving Systems and Higher-Order Equations Numerically
    5.4 Introduction to the Phase Plane
    5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models
    5.6 Coupled Mass-Spring Systems
    5.7 Electrical Systems
    5.8 Dynamical Systems, Poincaré Maps, and Chaos
    6. Theory of Higher-Order Linear Differential Equations
    6.1 Basic Theory of Linear Differential Equations
    6.2 Homogeneous Linear Equations with Constant Coefficients
    6.3 Undetermined Coefficients and the Annihilator Method
    6.4 Method of Variation of Parameters
    7. Laplace Transforms
    7.1 Introduction: A Mixing Problem
    7.2 Definition of the Laplace Transform
    7.3 Properties of the Laplace Transform
    7.4 Inverse Laplace Transform
    7.5 Solving Initial Value Problems
    7.6 Transforms of Discontinuous and Periodic Functions
    7.7 Convolution
    7.8 Impulses and the Dirac Delta Function
    7.9 Solving Linear Systems with Laplace Transforms
    8. Series Solutions of Differential Equations
    8.1 Introduction: The Taylor Polynomial Approximation
    8.2 Power Series and Analytic Functions
    8.3 Power Series Solutions to Linear Differential Equations
    8.4 Equations with Analytic Coefficients
    8.5 Cauchy-Euler (Equidimensional) Equations
    8.6 Method of Frobenius
    8.7 Finding a Second Linearly Independent Solution
    8.8 Special Functions
    9. Matrix Methods for Linear Systems
    9.1 Introduction
    9.2 Review 1: Linear Algebraic Equations
    9.3 Review 2: Matrices and Vectors
    9.4 Linear Systems in Normal Form
    9.5 Homogeneous Linear Systems with Constant Coefficients
    9.6 Complex Eigenvalues
    9.7 Nonhomogeneous Linear Systems
    9.8 The Matrix Exponential Function
    10. Partial Differential Equations
    10.1 Introduction: A Model for Heat Flow
    10.2 Method of Separation of Variables
    10.3 Fourier Series
    10.4 Fourier Cosine and Sine Series
    10.5 The Heat Equation
    10.6 The Wave Equation
    10.7 Laplace's Equation
    11. Eigenvalue Problems and Sturm-Liouville Equations
    11.1 Introduction: Heat Flow in a Nonuniform Wire
    11.2 Eigenvalues and Eigenfunctions
    11.3 Regular Sturm-Liouville Boundary Value Problems
    11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative
    11.5 Solution by Eigenfunction Expansion
    11.6 Green's Functions
    11.7 Singular Sturm-Liouville Boundary Value Problems.
    11.8 Oscillation and Comparison Theory
    12. Stability of Autonomous Systems
    12.1 Introduction: Competing Species
    12.2 Linear Systems in the Plane
    12.3 Almost Linear Systems
    12.4 Energy Methods
    12.5 Lyapunov's Direct Method
    12.6 Limit Cycles and Periodic Solutions
    12.7 Stability of Higher-Dimensional Systems
    13. Existence and Uniqueness Theory
    13.1 Introduction: Successive Approximations
    13.2 Picard's Existence and Uniqueness Theorem
    13.3 Existence of Solutions of Linear Equations
    13.4 Continuous Dependence of Solutions
    Appendices
    A. Review of Integration Techniques
    B. Newton's Method
    C. Simpson's Rule
    D. Cramer's Rule
    E. Method of Least Squares
    F. Runge-Kutta Procedure for n Equations
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