## Description

Fundamentals of Differential Equations presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. This flexible text allows instructors to adapt to various course emphases (theory, methodology, applications, and numerical methods) and to use commercially available computer software. For the first time, MyLab Math is available for this text, providing online homework with immediate feedback, the complete eText, and more.

Note that a longer version of this text, entitled Fundamentals of Differential Equations and Boundary Value Problems, 7th Edition, contains enough material for a two-semester course. This longer text consists of the main text plus three additional chapters (Eigenvalue Problems and SturmLiouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).

Also available with MyLab Math MyLab Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and pursue a personalized study plan that helps them absorb course material and understand difficult concepts.

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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 Functions
7.7. Transforms of Periodic and Power Functions
7.8. Convolution
7.9. Impulses and the Dirac Delta Function
7.10. 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

Appendix A. Newton's Method
Appendix B. Simpson's Rule
Appendix C. Cramer's Rule
Appendix D. Method of Least Squares
Appendix E. Runge-Kutta Procedure for n Equations
• Citation
• Full Title: Fundamentals of Differential Equations
• Author/s:
• ISBN-13: 9780321977069
• Edition: 9th Edition
• Publication date: 2017
• Topic: Math
• Subtopic: Differential Equations
• File Type: eBook
• Idioma: English

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