Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

• Preface 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 Curves 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 Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 6. Power Series Solutions and Special Functions6.1 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 7. Fourier Series: Basic Concepts7.1 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 Historical Note Anatomy of an Application Problems for Review and Discovery 8. Laplace Transforms8.0 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 Historical Note Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 10. Systems of First-Order Equations 10.1 Introductory Remarks 10.2 Linear Systems 10.3 Systems with Constant Coefficients 10.4 Nonlinear Systems Anatomy of an Application Problems for Review and Discovery 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 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 Liapunov’s Direct Method 12.6 Simple Critical Points of Nonlinear Systems 12.7 Nonlinear Mechanics: Conservative Systems 12.8 Periodic Solutions Historical Note Anatomy of an Application Problems for Review and Discovery 13. Qualitative Properties and Theoretical Aspects13.1 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 Anatomy of an Application Problems for Review and Discovery Appendix: Review of Linear Algebra BibliographyIndex

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

• Preface 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 Curves 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 Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 6. Power Series Solutions and Special Functions6.1 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 7. Fourier Series: Basic Concepts7.1 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 Historical Note Anatomy of an Application Problems for Review and Discovery 8. Laplace Transforms8.0 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 Historical Note Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 10. Systems of First-Order Equations 10.1 Introductory Remarks 10.2 Linear Systems 10.3 Systems with Constant Coefficients 10.4 Nonlinear Systems Anatomy of an Application Problems for Review and Discovery 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 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 Liapunov’s Direct Method 12.6 Simple Critical Points of Nonlinear Systems 12.7 Nonlinear Mechanics: Conservative Systems 12.8 Periodic Solutions Historical Note Anatomy of an Application Problems for Review and Discovery 13. Qualitative Properties and Theoretical Aspects13.1 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 Anatomy of an Application Problems for Review and Discovery Appendix: Review of Linear Algebra BibliographyIndex

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

• Preface 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 Curves 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 Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 6. Power Series Solutions and Special Functions6.1 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 7. Fourier Series: Basic Concepts7.1 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 Historical Note Anatomy of an Application Problems for Review and Discovery 8. Laplace Transforms8.0 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 Historical Note Anatomy of an Application Problems for Review and Discovery 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 Historical Note Anatomy of an Application Problems for Review and Discovery 10. Systems of First-Order Equations 10.1 Introductory Remarks 10.2 Linear Systems 10.3 Systems with Constant Coefficients 10.4 Nonlinear Systems Anatomy of an Application Problems for Review and Discovery 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 Historical Note Historical Note Anatomy of an Application Problems for Review and Discovery 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 Liapunov’s Direct Method 12.6 Simple Critical Points of Nonlinear Systems 12.7 Nonlinear Mechanics: Conservative Systems 12.8 Periodic Solutions Historical Note Anatomy of an Application Problems for Review and Discovery 13. Qualitative Properties and Theoretical Aspects13.1 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 Anatomy of an Application Problems for Review and Discovery Appendix: Review of Linear Algebra BibliographyIndex