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Differential Equations: A Modern Approach
This book is described in its preface as a second course in differential equations, but it would be more accurate to call it a course in elementary differential equations from an advanced standpoint. The prerequisites are modest: a good grasp of calculus and some knowledge of matrices and eigenvalues. The present work is a slightlycorrected 1975 reprint of the 1964 work published by Holt, Rinehart and Winston.
The book covers all the familiar solution methods for ordinary differential equations, but whenever possible works in the language of matrices and handles systems of equations rather than single equations. Much of the book takes a qualitative perspective, showing how to infer the behavior of solutions directly from the equations without finding an explicit solution. It is slanted to the theoretical side, and although it uses several equations from mathematical physics as examples, it does not explore the physical model or the consequences of the solution. It also omits any numerical work. The book has a large number of exercises, most of which are to solve particular equations but many of which ask for proofs.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

CHAPTER 1 LINEAR ALGEBRA

1.1 Introduction

1.2 Finite Dimensional Vector Spaces

1.3 Inner Products

1.4 Linear Transformations and Matrices

1.5 Computational Aspects of Matrix Operations

1.6 Eigenvalues and Eigenvectors

1.7 Norms of Matrices

1.8 Linear Vector Spaces

CHAPTER 2 LINEAR DIFFERENTIAL EQUATIONS

2.1 Introduction

2.2 FirstOrder Linear Differential Equations

2.3 Systems of Differential Equations

2.4 Systems of Linear Differential Equations

2.5 Homogeneous Systems of Linear Differential Equations with Constant Coefficients

2.6 The Laplace Transform

2.7 Inhomogeneous Systems of Linear Differential Equations

2.8 General Theory of Linear Differential Equations

CHAPTER 3 SERIES SOLUTIONS FOR LINEAR DIFFERENTIAL EQUATIONS

3.1 Differential Equations with Analytic Coefficients

3.2 Euler’s Differential Equations

3.3 Equations with Regular Singularities

3.4 Asymptotic Solutions

CHAPTER 4 BOUNDARY VALUE PROBLEMS

4.1 Introduction

4.2 Normal Forms

4.3 The Equation \(u'' + \lambda u = 0\)

4.4 The Equation \(y'' + [\lambda  q(z)] y = 0\)

4.5 Integral Equations

4.6 Function Spaces

4.7 Fourier Expansions

4.8 Generalizations

CHAPTER 5 LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

5.1 Linear Systems with Periodic Coefficients

5.2 Applications to SecondOrder Equations

CHAPTER 6 NONLINEAR DIFFERENTIAL EQUATIONS

6.1 Fundamental Existence Theorem

6.2 Series Solutions

6.3 Stability Theory

6.4 Lyapunov’s Direct Method

6.5 Differential Equations with Periodic Solutions

CHAPTER 7 TwoDIMENSIONAL SYSTEMS

7.1 Autonomous Systems

7.2 Limit Cycles

7.3 The Poincaré Index

7.4 Perturbation Theory of SecondOrder Systems

BIBLIOGRAPHY

INDEX
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