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Full Description
Fundamentals of Differential Equations 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, course emphasis (theory, methodology, applications, and numerical methods), and in using commercially available computer software.Fundamentals of Differential Equations, Eighth Edition is suitable for a one-semester sophomore- or junior-level course. Fundamentals of Differential Equations with Boundary Value Problems, Sixth Edition, contains enough material for a two-semester course that covers and builds on boundary value problems. The Boundary Value Problems version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).
Contents
1. Introduction1.1 Background1.2 Solutions and Initial Value Problems1.3 Direction Fields1.4 The Approximation Method of EulerChapter SummaryTechnical Writing ExercisesGroup Projects for Chapter 1A. Taylor Series MethodB. Picard's MethodC. The Phase Line2. First-Order Differential Equations2.1 Introduction: Motion of a Falling Body2.2 Separable Equations2.3 Linear Equations2.4 Exact Equations2.5 Special Integrating Factors2.6 Substitutions and TransformationsChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 2A. Oil Spill in a CanalB. Differential Equations in Clinical MedicineC. Torricelli's Law of Fluid FlowD. The Snowplow ProblemE. Two SnowplowsF. Clairaut Equations and Singular SolutionsG. Multiple Solutions of a First-Order Initial Value ProblemH. Utility Functions and Risk AversionI. Designing a Solar CollectorJ. Asymptotic Behavior of Solutions to Linear Equations3. Mathematical Models and Numerical Methods Involving First Order Equations3.1 Mathematical Modeling3.2 Compartmental Analysis3.3 Heating and Cooling of Buildings3.4 Newtonian Mechanics3.5 Electrical Circuits3.6 Improved Euler's Method3.7 Higher-Order Numerical Methods: Taylor and Runge-KuttaGroup Projects for Chapter 3A. Dynamics of HIV InfectionB. AquacultureC. Curve of PursuitD. Aircraft Guidance in a CrosswindE. Feedback and the Op AmpF. Bang-Bang ControlsG. Market Equilibrium: Stability and Time PathsH. Stability of Numerical MethodsI. Period Doubling and Chaos4. Linear Second-Order Equations4.1 Introduction: The Mass-Spring Oscillator4.2 Homogeneous Linear Equations: The General Solution4.3 Auxiliary Equations with Complex Roots4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients4.5 The Superposition Principle and Undetermined Coefficients Revisited4.6 Variation of Parameters4.7 Variable-Coefficient Equations4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations4.9 A Closer Look at Free Mechanical Vibrations4.10 A Closer Look at Forced Mechanical VibrationsChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 4A. Nonlinear Equations Solvable by First-Order TechniquesB. Apollo ReentryC. Simple PendulumD. Linearization of Nonlinear ProblemsE. Convolution MethodF. Undetermined Coefficients Using Complex ArithmeticG. Asymptotic Behavior of Solutions5. Introduction to Systems and Phase Plane Analysis5.1 Interconnected Fluid Tanks5.2 Elimination Method for Systems with Constant Coefficients5.3 Solving Systems and Higher-Order Equations Numerically5.4 Introduction to the Phase Plane5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models5.6 Coupled Mass-Spring Systems5.7 Electrical Systems5.8 Dynamical Systems, Poincare Maps, and ChaosChapter SummaryReview ProblemsGroup Projects for Chapter 5A. Designing a Landing System for Interplanetary TravelB. Spread of Staph Infections in Hospitals-Part 1C. Things That BobD. Hamiltonian SystemsE. Cleaning Up the Great Lakes6. Theory of Higher-Order Linear Differential Equations6.1 Basic Theory of Linear Differential Equations6.2 Homogeneous Linear Equations with Constant Coefficients6.3 Undetermined Coefficients and the Annihilator Method6.4 Method of Variation of ParametersChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 6A. Computer Algebra Systems and Exponential ShiftB. Justifying the Method of Undetermined CoefficientsC. Transverse Vibrations of a Beam7. Laplace Transforms7.1 Introduction: A Mixing Problem7.2 Definition of the Laplace Transform7.3 Properties of the Laplace Transform7.4 Inverse Laplace Transform7.5 Solving Initial Value Problems7.6 Transforms of Discontinuous and Periodic Functions7.7 Convolution7.8 Impulses and the Dirac Delta Function7.9 Solving Linear Systems with Laplace TransformsChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 7A. Duhamel's FormulasB. Frequency Response ModelingC. Determining System Parameters8. Series Solutions of Differential Equations8.1 Introduction: The Taylor Polynomial Approximation8.2 Power Series and Analytic Functions8.3 Power Series Solutions to Linear Differential Equations8.4 Equations with Analytic Coefficients8.5 Cauchy-Euler (Equidimensional) Equations8.6 Method of Frobenius8.7 Finding a Second Linearly Independent Solution8.8 Special FunctionsChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 8A. Alphabetization AlgorithmsB. Spherically Symmetric Solutions to Shroedinger's Equation for the Hydrogen AtomC. Airy's EquationD. Buckling of a TowerE. Aging Spring and Bessel Functions9. Matrix Methods for Linear Systems9.1 Introduction9.2 Review 1: Linear Algebraic Equations9.3 Review 2: Matrices and Vectors9.4 Linear Systems in Normal Form9.5 Homogeneous Linear Systems with Constant Coefficients9.6 Complex Eigenvalues9.7 Nonhomogeneous Linear Systems9.8 The Matrix Exponential FunctionChapter SummaryReview ProblemsTechnical Writing ExercisesGroup Projects for Chapter 9A. Uncoupling Normal SystemsB. Matrix Laplace Transform MethodC. Undamped Second-Order SystemsD. Undetermined Coefficients for System Forced by Homogeneous10. Partial Differential Equations10.1 Introduction: A Model for Heat Flow10.2 Method of Separation of Variables10.3 Fourier Series10.4 Fourier Cosine and Sine Series10.5 The Heat Equation10.6 The Wave Equation10.7 Laplace's EquationChapter SummaryTechnical Writing ExercisesGroup Projects for Chapter 10A. Steady-State Temperature Distribution in a Circular CylinderB. A Laplace Transform Solution of the Wave EquationC. Green's FunctionD. Numerical Method for u=f on a RectangleAppendicesA. Newton's MethodB. Simpson's RuleC. Cramer's RuleD. Method of Least SquaresE. Runge-Kutta Procedure for n EquationsAnswers to Odd-Numbered ProblemsIndex
