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Index
Cover
Contents
Title Page
Copyright
Preface
CHAPTER 1: FIRST-ORDER DIFFERENTIAL EQUATIONS
1.1 MOTIVATION AND OVERVIEW
1.2 LINEAR FIRST-ORDER EQUATIONS
1.3 APPLICATIONS OF LINEAR FIRST-ORDER EQUATIONS
1.4 NONLINEAR FIRST-ORDER EQUATIONS THAT ARE SEPARABLE
1.5 EXISTENCE AND UNIQUENESS
1.6 APPLICATIONS OF NONLINEAR FIRST-ORDER EQUATIONS
1.7 EXACT EQUATIONS AND EQUATIONS THAT CAN BE MADE EXACT
1.8 SOLUTION BY SUBSTITUTION
1.9 NUMERICAL SOLUTION BY EULER’S METHOD
CHAPTER 1 REVIEW
CHAPTER 2: HIGHER-ORDER LINEAR EQUATIONS
2.1 LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER
2.2 CONSTANT-COEFFICIENT EQUATIONS
2.3 COMPLEX ROOTS
2.4 LINEAR INDEPENDENCE; EXISTENCE, UNIQUENESS, GENERAL SOLUTION
2.5 REDUCTION OF ORDER
2.6 CAUCHY–EULER EQUATIONS
2.7 THE GENERAL THEORY FOR HIGHER-ORDER EQUATIONS
2.8 NONHOMOGENEOUS EQUATIONS
2.9 PARTICULAR SOLUTION BY UNDETERMINED COEFFICIENTS
2.10 PARTICULAR SOLUTION BY VARIATION OF PARAMETERS
CHAPTER 2 REVIEW
CHAPTER 3: APPLICATIONS OF HIGHER-ORDER EQUATIONS
3.1 INTRODUCTION
3.2 LINEAR HARMONIC OSCILLATOR; FREE OSCILLATION
3.3 FREE OSCILLATION WITH DAMPING
3.4 FORCED OSCILLATION
3.5 STEADY-STATE DIFFUSION; A BOUNDARY VALUE PROBLEM
3.6 INTRODUCTION TO THE EIGENVALUE PROBLEM; COLUMN BUCKLING
CHAPTER 3 REVIEW
CHAPTER 4: SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
4.1 INTRODUCTION, AND SOLUTION BY ELIMINATION
4.2 APPLICATION TO COUPLED OSCILLATORS
4.3 N-SPACE AND MATRICES
4.4 LINEAR DEPENDENCE AND INDEPENDENCE OF VECTORS
4.5 EXISTENCE, UNIQUENESS, AND GENERAL SOLUTION
4.6 MATRIX EIGENVALUE PROBLEM
4.7 HOMOGENEOUS SYSTEMS WITH CONSTANT COEFFICIENTS
4.8 DOT PRODUCT AND ADDITIONAL MATRIX ALGEBRA
4.9 EXPLICIT SOLUTION OF x′ = Ax AND THE MATRIX EXPONENTIAL FUNCTION
4.10 NONHOMOGENEOUS SYSTEMS
CHAPTER 4 REVIEW
CHAPTER 5: LAPLACE TRANSFORM
5.1 INTRODUCTION
5.2 THE TRANSFORM AND ITS INVERSE
5.3 APPLICATION TO THE SOLUTION OF DIFFERENTIAL EQUATIONS
5.4 DISCONTINUOUS FORCING FUNCTIONS; HEAVISIDE STEP FUNCTION
5.5 CONVOLUTION
5.6 IMPULSIVE FORCING FUNCTIONS; DIRAC DELTA FUNCTION
CHAPTER 5 REVIEW
CHAPTER 6: SERIES SOLUTIONS
6.1 INTRODUCTION
6.2 POWER SERIES AND TAYLOR SERIES
6.3 POWER SERIES SOLUTION ABOUT A REGULAR POINT
6.4 LEGENDRE AND BESSEL EQUATIONS
6.5 THE METHOD OF FROBENIUS
CHAPTER 6 REVIEW
CHAPTER 7: SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS
7.1 INTRODUCTION
7.2 THE PHASE PLANE
7.3 LINEAR SYSTEMS
7.4 NONLINEAR SYSTEMS
7.5 LIMIT CYCLES
7.6 NUMERICAL SOLUTION OF SYSTEMS BY EULER’S METHOD
CHAPTER 7 REVIEW
APPENDIX A: REVIEW OF PARTIAL FRACTION EXPANSIONS
APPENDIX B: REVIEW OF DETERMINANTS
APPENDIX C: REVIEW OF GAUSS ELIMINATION
APPENDIX D: REVIEW OF COMPLEX NUMBERS AND THE COMPLEX PLANE
ANSWERS TO EXERCISES
INDEX
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