## Description

We wrote this book to provide a concrete and readable text for the traditional course in elementary differential equations that science, engineering, and mathematics students take following calculus. Our treatment is shaped throughout by the goal of a flexible exposition that students will find accessible, attractive, and interesting.

The applications of differential equations have played a singular role in the historical development of the subject, and whole areas of study exist mainly because of their applications. We therefore want our students to learn first to solve those differential equations that enjoy the most frequent and interesting applications. Hence we make consistent use of appealing applications to motivate and illustrate the standard elementary techniques of solution of differential equations.

The first course in differential equations should also be a window on the world of mathematics. While it is neither feasible nor desirable to include proofs of the fundamental existence and uniqueness theorems along the way in an elementary course, students need to see precise and clear-cut statements of these theorems, and to understand their role in the subject. We include appropriate existence and uniqueness proofs in the Appendix, and occasionally refer to them in the main body of the text.

The list of introductory topics in differential equations is quite standard, and a glance at our chapter titles will reveal no major surprises, though in the fine structure we have attempted to add a bit of zest here and there. In most chapters the principal ideas of the topic are introduced in the first few sections of the chapter, and the remaining sections are devoted to extensions and applications. Hence the instructor has a wide range of choice regarding breadth and depth of coverage.

At various points our approach reflects the widespread use of computer programs for the numerical solution of differential equations. Nevertheless, we continue to believe that the traditional elementary analytical methods of solution are important for students to learn. One reason is that effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques; the construction of a realistic numerical model often is based on the study of a simpler analytical model.

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• 1. First-Order Differential Equations
1.1 Differential Equations and Mathematical Models
1.2 Integrals as General and Particular Solutions
1.3 Slope Fields and Solution Curves
1.4 Separable Equations and Applications
1.5 Linear First-Order Equations
1.6 Substitution Methods and Exact Equations

2. Mathematical Models and Numerical Methods
2.1 Population Models
2.2 Equilibrium Solutions and Stability
2.3 AccelerationVelocity Models
2.4 Numerical Approximation: Eulers Method
2.5 A Closer Look at the Euler Method
2.6 The RungeKutta Method

3. Linear Equations of Higher Order
3.1 Introduction: Second-Order Linear Equations
3.2 General Solutions of Linear Equations
3.3 Homogeneous Equations with Constant Coefficients
3.4 Mechanical Vibrations
3.5 Nonhomogeneous Equations and Undetermined Coefficients
3.6 Forced Oscillations and Resonance
3.7 Electrical Circuits
3.8 Endpoint Problems and Eigenvalues

4. Introduction to Systems of Differential Equations
4.1 First-Order Systems and Applications
4.2 The Method of Elimination
4.3 Numerical Methods for Systems

5. Linear Systems of Differential Equations
5.1 Matrices and Linear Systems
5.2 The Eigenvalue Method for Homogeneous Systems
5.3 A Gallery of Solution Curves of Linear Systems
5.4 Second-Order Systems and Mechanical Applications
5.5 Multiple Eigenvalue Solutions
5.6 Matrix Exponentials and Linear Systems
5.7 Nonhomogeneous Linear Systems

6. Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane
6.2 Linear and Almost Linear Systems
6.3 Ecological Models: Predators and Competitors
6.4 Nonlinear Mechanical Systems
6.5 Chaos in Dynamical Systems

7. Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms
7.2 Transformation of Initial Value Problems
7.3 Translation and Partial Fractions
7.4 Derivatives, Integrals, and Products of Transforms
7.5 Periodic and Piecewise Continuous Input Functions
7.6 Impulses and Delta Functions

8. Power Series Methods
8.1 Introduction and Review of PowerSeries
8.2 Series Solutions Near Ordinary Points
8.3 Regular Singular Points
8.4 Method of Frobenius: The Exceptional Cases
8.5 Bessels Equation
8.6 Applications of Bessel Functions

9. Fourier Series Methods and Partial Differential Equations
9.1 Periodic Functions and Trigonometric Series
9.2 General Fourier Series and Convergence
9.3 Fourier Sine and Cosine Series
9.4 Applications of Fourier Series
9.5 Heat Conduction and Separation of Variables
9.6 Vibrating Strings and the One-Dimensional Wave Equation
9.7 Steady-State Temperature and Laplaces Equation

10. Eigenvalue Methods and Boundary Value Problems
10.1 SturmLiouville Problems and Eigenfunction Expansions
10.2 Applications of Eigenfunction Series
10.3 Steady Periodic Solutions and Natural Frequencies
10.4 Cylindrical Coordinate Problems
10.5 Higher-Dimensional Phenomena
• Citation
• Full Title: Elementary Differential Equations with Boundary Value Problems
• Author/s:
• ISBN-10: 0131457748
• ISBN-13: 9780131457744
• Edition: 5th Edition
• Publication date: 2003
• Topic: Math
• Subtopic: Differential Equations
• File Type: Solution Manual

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