Differential equations : theory, technique, and practice

Differential equations theory, technique, and practice

Differential equations is one of the oldest subjects in modern mathematics. It was not long after Newton and Leibniz invented the calculus that Bernoulli and Euler and others began to consider the heat equation and the wave equation of mathematical physics. Newton himself solved differential equatio...

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Detalles Bibliográficos
Otros Autores: Krantz, Steven G. 1951- author (author), Nolan, Carl A., author
Formato: Libro electrónico
Idioma:Inglés
Publicado: Boca Raton : CRC Press [2022]
Edición:Third edition
Colección:Textbooks in mathematics (Boca Raton, Fla.)
Materias:
Differential equations.
Ver en Biblioteca Universitat Ramon Llull:https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009757919006719
Tabla de Contenidos:
  • Cover
  • Half Title
  • Series Page
  • Title Page
  • Copyright Page
  • Dedication
  • Contents
  • Preface
  • Author
  • 1. What is a Differential Equation?
  • 1.1. Introductory Remarks
  • 1.2. A Taste of Ordinary Differential Equations
  • 1.3. The Nature of Solutions
  • 2. Solving First-Order Equations
  • 2.1. Separable Equations
  • 2.2. First-Order Linear Equations
  • 2.3. Exact Equations
  • 2.4. Orthogonal Trajectories and Families
  • 2.5. Homogeneous Equations
  • 2.6. Integrating Factors
  • 2.7. Reduction of Order
  • 2.7.1. Dependent Variable Missing
  • 2.7.2. Independent Variable Missing
  • 3. Some Applications of the First-Order Theory
  • 3.1. The Hanging Chain and Pursuit Curves
  • 3.1.1. The Hanging Chain
  • 3.1.2. Pursuit Curves
  • 3.2. Electrical Circuits
  • 4. Second-Order Linear Equations
  • 4.1. Second-Order Linear Equations with Constant Coefficients
  • 4.2. The Method of Undetermined Coefficients
  • 4.3. The Method of Variation of Parameters
  • 4.4. The Use of a Known Solution to Find Another
  • 4.5. Higher-Order Equations
  • 5. Applications of the Second-Order Theory
  • 5.1. Vibrations and Oscillations
  • 5.1.1. Undamped Simple Harmonic Motion
  • 5.1.2. Damped Vibrations
  • 5.1.3. Forced Vibrations
  • 5.1.4. A Few Remarks about Electricity
  • 5.2. Newton's Law of Gravitation and Kepler's Laws
  • 5.2.1. Kepler's Second Law
  • 5.2.2. Kepler's First Law
  • 5.2.3. Kepler's Third Law
  • 6. Power Series Solutions and Special Functions
  • 6.1. Introduction and Review of Power Series
  • 6.1.1. Review of Power Series
  • 6.2. Series Solutions of First-Order Equations
  • 6.3. Ordinary Points
  • 6.4. Regular Singular Points
  • 6.5. More on Regular Singular Points
  • 7. Fourier Series: Basic Concepts
  • 7.1. Fourier Coefficients
  • 7.2. Some Remarks about Convergence
  • 7.3. Even and Odd Functions: Cosine and Sine Series.
  • 7.4. Fourier Series on Arbitrary Intervals
  • 7.5. Orthogonal Functions
  • 8. Laplace Transforms
  • 8.0. Introduction
  • 8.1. Applications to Differential Equations
  • 8.2. Derivatives and Integrals
  • 8.3. Convolutions
  • 8.3.1. Abel's Mechanics Problem
  • 8.4. The Unit Step and Impulse Functions
  • 9. The Calculus of Variations
  • 9.1. Introductory Remarks
  • 9.2. Euler's Equation
  • 9.3. Isoperimetric Problems and the Like
  • 9.3.1. Lagrange Multipliers
  • 9.3.2. Integral Side Conditions
  • 9.3.3. Finite Side Conditions
  • 10. Systems of First-Order Equations
  • 10.1. Introductory Remarks
  • 10.2. Linear Systems
  • 10.3. Systems with Constant Coefficients
  • 10.4. Nonlinear Systems
  • 11. Partial Differential Equations and Boundary Value Problems
  • 11.1. Introduction and Historical Remarks
  • 11.2 Eigenvalues and the Vibrating String
  • 11.2.1. Boundary Value Problems
  • 11.2.2. Derivation of the Wave Equation
  • 11.2.3. Solution of the Wave Equation
  • 11.3. The Heat Equation
  • 11.4. The Dirichlet Problem for a Disc
  • 11.4.1. The Poisson Integral
  • 11.5. Sturm-Liouville Problems
  • 12. The Nonlinear Theory
  • 12.1. Some Motivating Examples
  • 12.2. Specializing Down
  • 12.3. Types of Critical Points: Stability
  • 12.4. Critical Points and Stability
  • 12.5. Stability by Lyapunov's Direct Method
  • 12.6. Simple Critical Points of Nonlinear Systems
  • 12.7. Nonlinear Mechanics: Conservative Systems
  • 12.8. Periodic Solutions
  • 13. Qualitative Properties and Theoretical Aspects
  • 13.1. A Bit of Theory
  • 13.2. Picard's Existence and Uniqueness Theorem
  • 13.2.1. The Form of a Differential Equation
  • 13.2.2. Picard's Iteration Technique
  • 13.2.3. Some Illustrative Examples
  • 13.2.4. Estimation of the Picard Iterates
  • 13.3. Oscillations and the Sturm Separation Theorem
  • 13.4. The Sturm Comparison Theorem.
  • APPENDIX: Review of Linear Algebra
  • Bibliography
  • Index.

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