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Hardcover ISBN:  9781470451738 
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MAA Member Price:  $123.00 $93.38 
AMS Member Price:  $123.00 $93.38 
Hardcover ISBN:  9781470451738 
Product Code:  TEXT/54 
List Price:  $85.00 
MAA Member Price:  $63.75 
AMS Member Price:  $63.75 
eBook ISBN:  9781470453640 
Product Code:  TEXT/54.E 
List Price:  $79.00 
MAA Member Price:  $59.25 
AMS Member Price:  $59.25 
Hardcover ISBN:  9781470451738 
eBook ISBN:  9781470453640 
Product Code:  TEXT/54.B 
List Price:  $164.00 $124.50 
MAA Member Price:  $123.00 $93.38 
AMS Member Price:  $123.00 $93.38 

Book DetailsAMS/MAA TextbooksVolume: 54; 2019; 399 ppMSC: Primary 34; 35; 44; 42
Lectures on Differential Equations provides a clear and concise presentation of differential equations for undergraduates and beginning graduate students. There is more than enough material here for a yearlong course. In fact, the text developed from the author's notes for three courses: the undergraduate introduction to ordinary differential equations, the undergraduate course in Fourier analysis and partial differential equations, and a first graduate course in differential equations. The first four chapters cover the classical syllabus for the undergraduate ODE course leavened by a modern awareness of computing and qualitative methods. The next two chapters contain a welldeveloped exposition of linear and nonlinear systems with a similarly fresh approach. The final two chapters cover boundary value problems, Fourier analysis, and the elementary theory of PDEs.
The author makes a concerted effort to use plain language and to always start from a simple example or application. The presentation should appeal to, and be readable by, students, especially students in engineering and science. Without being excessively theoretical, the book does address a number of unusual topics: Massera's theorem, Lyapunov's inequality, the isoperimetric inequality, numerical solutions of nonlinear boundary value problems, and more. There are also some new approaches to standard topics including a rethought presentation of series solutions and a nonstandard, but more intuitive, proof of the existence and uniqueness theorem. The collection of problems is especially rich and contains many very challenging exercises.
Philip Korman is professor of mathematics at the University of Cincinnati. He is the author of over one hundred research articles in differential equations and the monograph Global Solution Curves for Semilinear Elliptic Equations. Korman has served on the editorial boards of Communications on Applied Nonlinear Analysis, Electronic Journal of Differential Equations, SIAM Review, an\ d Differential Equations and Applications.
Ancillaries:
ReadershipUndergraduate and graduate students interested in differential equations.

Table of Contents

Cover

Title page

Copyright

Contents

Introduction

Chapter 1. FirstOrder Equations

1.1. Integration by GuessandCheck

1.2. FirstOrder Linear Equations

1.2.1. The Integrating Factor

1.3. Separable Equations

1.3.1. Problems

1.4. Some Special Equations

1.4.1. Homogeneous Equations

1.4.2. The Logistic Population Model

1.4.3. Bernoulli’s Equations

1.4.4*. Riccati’s Equations

1.4.5*. Parametric Integration

1.4.6. Some Applications

1.5. Exact Equations

1.6. Existence and Uniqueness of Solution

1.7. Numerical Solution by Euler’s Method

1.7.1. Problems

1.8*. The Existence and Uniqueness Theorem

1.8.1. Problems

Chapter 2. SecondOrder Equations

2.1. Special SecondOrder Equations

2.1.1. 𝑦 Is Not Present in the Equation

2.1.2. 𝑥 Is Not Present in the Equation

2.1.3*. The Trajectory of Pursuit

2.2. Linear Homogeneous Equations with Constant Coefficients

2.2.1. The Characteristic Equation Has Two Distinct Real Roots

2.2.2. The Characteristic Equation Has Only One (Repeated) Real Root

2.3. The Characteristic Equation Has Two Complex Conjugate Roots

2.3.1. Euler’s Formula

2.3.2. The General Solution

2.3.3. Problems

2.4. Linear SecondOrder Equations with Variable Coefficients

2.5. Some Applications of the Theory

2.5.1. The Hyperbolic Sine and Cosine Functions

2.5.2. Different Ways to Write the General Solution

2.5.3. Finding the Second Solution

2.5.4. Problems

2.6. Nonhomogeneous Equations

2.7. More on Guessing of 𝑌(𝑡)

2.8. The Method of Variation of Parameters

2.9. The Convolution Integral

2.9.1. Differentiation of Integrals

2.9.2. Yet Another Way to Compute a Particular Solution

2.10. Applications of SecondOrder Equations

2.10.1. Vibrating Spring

2.10.2. Problems

2.10.3. A Meteor Approaching the Earth

2.10.4. Damped Oscillations

2.11. Further Applications

2.11.1. Forced and Damped Oscillations

2.11.2. An Example of a Discontinuous Forcing Term

2.11.3. Oscillations of a Pendulum

2.11.4. Sympathetic Oscillations

2.12. Oscillations of a Spring Subject to a Periodic Force

2.12.1. The Fourier Series

2.12.2. Vibrations of a Spring Subject to a Periodic Force

2.13. Euler’s Equation

2.14. Linear Equations of Order Higher Than Two

2.14.1. The Polar Form of Complex Numbers

2.14.2. Linear Homogeneous Equations

2.14.3. Nonhomogeneous Equations

2.14.4. Problems

2.15. Oscillation and Comparison Theorems

Chapter 3. Using Infinite Series to Solve Differential Equations

3.1. Series Solution Near a Regular Point

3.1.1. Maclauren and Taylor Series

3.1.2. A Toy Problem

3.1.3. Using Series When Other Methods Fail

3.2. Solution Near a Mildly Singular Point

3.2.1*. Derivation of 𝐽₀(𝑥) by Differentiation of the Equation

3.3. Moderately Singular Equations

3.3.1. Problems

Chapter 4. The Laplace Transform

4.1. The Laplace Transform and Its Inverse

4.1.1. Review of Improper Integrals

4.1.2. The Laplace Transform

4.1.3. The Inverse Laplace Transform

4.2. Solving the Initial Value Problems

4.2.1. Step Functions

4.3. The Delta Function and Impulse Forces

4.4. Convolution and the Tautochrone Curve

4.4.1. The Tautochrone curve

4.5. Distributions

4.5.1. Problems

Chapter 5. Linear Systems ofDifferential Equations

5.1. The Case of Real Eigenvalues

5.1.1. Review of Vectors and Matrices

5.1.2. Linear FirstOrder Systems with Constant Coefficients

5.2. A Pair of Complex Conjugate Eigenvalues

5.2.1. ComplexValued and RealValued Solutions

5.2.2. The General Solution

5.2.3. Nonhomogeneous Systems

5.3. The Exponential of a Matrix

5.3.1. Problems

5.4. Floquet Theory and Massera’s Theorem

5.5. Solutions of Planar Systems Near the Origin

5.5.1. Linearization and the HartmanGrobman Theorem

5.5.2. Phase Plane and the Prüfer Transformation

5.5.3. Problems

5.6. Controllability and Observability

5.6.1. The CayleyHamilton Theorem

5.6.2. Controllability of Linear Systems

5.6.3. Observability

5.6.4. Problems

Chapter 6. Nonlinear Systems

6.1. The PredatorPrey Interaction

6.2. Competing Species

6.3. An Application to Epidemiology

6.4. Lyapunov’s Stability

6.4.1. Stable Systems

6.5. Limit Cycles

6.5.1. PoincaréBendixson Theorem

6.5.2. DulacBendixson Criterion

6.6. Periodic Population Models

6.6.1. Problems

Chapter 7. The Fourier Series and Boundary Value Problems

7.1. The Fourier Series for Functions of an Arbitrary Period

7.1.1. Even and Odd Functions

7.1.2. Further Examples and the Convergence Theorem

7.2. The Fourier Cosine and the Fourier SineSeries

7.3. TwoPoint Boundary Value Problems

7.3.1. Problems

7.4. The Heat Equation and the Method of Separation of Variables

7.5. Laplace’s Equation

7.6. The Wave Equation

7.6.1. Nonhomogeneous Problems

7.6.2. Problems

7.7. Calculating Earth’s Temperature and Queen Dido’s Problem

7.7.1. The Complex Form of the Fourier Series

7.7.2. The Temperatures Inside the Earth and Wine Cellars

7.7.3. The Isoperimetric Inequality

7.8. Laplace’s Equation on Circular Domains

7.9. SturmLiouville Problems

7.9.1. The FourierBessel Series

7.9.2. Cooling of a Cylindrical Tank

7.9.3. Cooling of a Rectangular Bar

7.10. Green’s Function

7.10.1. Problems

7.11. The Fourier Transform

7.12. Problems on Infinite Domains

7.12.1. Evaluation of Some Integrals

7.12.2. The Heat Equation for ∞<𝑥<∞

7.12.3. SteadyState Temperatures for the Upper HalfPlane

7.12.4. Using the Laplace Transform for a SemiInfinite String

7.12.5. Problems

Chapter 8. Elementary Theory of PDE

8.1. Wave Equation: Vibrations of an InfiniteString

8.2. SemiInfinite String: Reflection of Waves

8.3. Bounded String: Multiple Reflections

8.4. Neumann Boundary Conditions

8.5. Nonhomogeneous Wave Equation

8.5.1. Problems

8.6. FirstOrder Linear Equations

8.6.1. Problems

8.7. Laplace’s Equation: Poisson’s Integral Formula

8.8. Some Properties of Harmonic Functions

8.9. The Maximum Principle

8.10. The Maximum Principle for the Heat Equation

8.10.1. Uniqueness on an Infinite Interval

8.11. Dirichlet’s Principle

8.12. Classification Theory for Two Variables

8.12.1. Problems

Chapter 9. Numerical Computations

9.1. The Capabilities of Software Systems, Like Mathematica

9.2. Solving Boundary Value Problems

9.3. Solving Nonlinear Boundary Value Problems

9.4. Direction Fields

Appendix

A.1. The Chain Rule and Its Descendants

A.2. Partial Fractions

A.3. Eigenvalues and Eigenvectors

A.4. Matrix Functions and the Norm

Bibliography

Index

Back Cover


Additional Material

Reviews

In this introduction to differential equations, the author emphasizes that he wanted to write a book that students would like to read. While theoretical results are mentioned, stated, and occasionally proved throughout the book, the focus is on solving differential equations and understanding properties of their solutions.
Bill Satzer, MAA Reviews


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseInstructor's Solutions Manual – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Lectures on Differential Equations provides a clear and concise presentation of differential equations for undergraduates and beginning graduate students. There is more than enough material here for a yearlong course. In fact, the text developed from the author's notes for three courses: the undergraduate introduction to ordinary differential equations, the undergraduate course in Fourier analysis and partial differential equations, and a first graduate course in differential equations. The first four chapters cover the classical syllabus for the undergraduate ODE course leavened by a modern awareness of computing and qualitative methods. The next two chapters contain a welldeveloped exposition of linear and nonlinear systems with a similarly fresh approach. The final two chapters cover boundary value problems, Fourier analysis, and the elementary theory of PDEs.
The author makes a concerted effort to use plain language and to always start from a simple example or application. The presentation should appeal to, and be readable by, students, especially students in engineering and science. Without being excessively theoretical, the book does address a number of unusual topics: Massera's theorem, Lyapunov's inequality, the isoperimetric inequality, numerical solutions of nonlinear boundary value problems, and more. There are also some new approaches to standard topics including a rethought presentation of series solutions and a nonstandard, but more intuitive, proof of the existence and uniqueness theorem. The collection of problems is especially rich and contains many very challenging exercises.
Philip Korman is professor of mathematics at the University of Cincinnati. He is the author of over one hundred research articles in differential equations and the monograph Global Solution Curves for Semilinear Elliptic Equations. Korman has served on the editorial boards of Communications on Applied Nonlinear Analysis, Electronic Journal of Differential Equations, SIAM Review, an\ d Differential Equations and Applications.
Ancillaries:
Undergraduate and graduate students interested in differential equations.

Cover

Title page

Copyright

Contents

Introduction

Chapter 1. FirstOrder Equations

1.1. Integration by GuessandCheck

1.2. FirstOrder Linear Equations

1.2.1. The Integrating Factor

1.3. Separable Equations

1.3.1. Problems

1.4. Some Special Equations

1.4.1. Homogeneous Equations

1.4.2. The Logistic Population Model

1.4.3. Bernoulli’s Equations

1.4.4*. Riccati’s Equations

1.4.5*. Parametric Integration

1.4.6. Some Applications

1.5. Exact Equations

1.6. Existence and Uniqueness of Solution

1.7. Numerical Solution by Euler’s Method

1.7.1. Problems

1.8*. The Existence and Uniqueness Theorem

1.8.1. Problems

Chapter 2. SecondOrder Equations

2.1. Special SecondOrder Equations

2.1.1. 𝑦 Is Not Present in the Equation

2.1.2. 𝑥 Is Not Present in the Equation

2.1.3*. The Trajectory of Pursuit

2.2. Linear Homogeneous Equations with Constant Coefficients

2.2.1. The Characteristic Equation Has Two Distinct Real Roots

2.2.2. The Characteristic Equation Has Only One (Repeated) Real Root

2.3. The Characteristic Equation Has Two Complex Conjugate Roots

2.3.1. Euler’s Formula

2.3.2. The General Solution

2.3.3. Problems

2.4. Linear SecondOrder Equations with Variable Coefficients

2.5. Some Applications of the Theory

2.5.1. The Hyperbolic Sine and Cosine Functions

2.5.2. Different Ways to Write the General Solution

2.5.3. Finding the Second Solution

2.5.4. Problems

2.6. Nonhomogeneous Equations

2.7. More on Guessing of 𝑌(𝑡)

2.8. The Method of Variation of Parameters

2.9. The Convolution Integral

2.9.1. Differentiation of Integrals

2.9.2. Yet Another Way to Compute a Particular Solution

2.10. Applications of SecondOrder Equations

2.10.1. Vibrating Spring

2.10.2. Problems

2.10.3. A Meteor Approaching the Earth

2.10.4. Damped Oscillations

2.11. Further Applications

2.11.1. Forced and Damped Oscillations

2.11.2. An Example of a Discontinuous Forcing Term

2.11.3. Oscillations of a Pendulum

2.11.4. Sympathetic Oscillations

2.12. Oscillations of a Spring Subject to a Periodic Force

2.12.1. The Fourier Series

2.12.2. Vibrations of a Spring Subject to a Periodic Force

2.13. Euler’s Equation

2.14. Linear Equations of Order Higher Than Two

2.14.1. The Polar Form of Complex Numbers

2.14.2. Linear Homogeneous Equations

2.14.3. Nonhomogeneous Equations

2.14.4. Problems

2.15. Oscillation and Comparison Theorems

Chapter 3. Using Infinite Series to Solve Differential Equations

3.1. Series Solution Near a Regular Point

3.1.1. Maclauren and Taylor Series

3.1.2. A Toy Problem

3.1.3. Using Series When Other Methods Fail

3.2. Solution Near a Mildly Singular Point

3.2.1*. Derivation of 𝐽₀(𝑥) by Differentiation of the Equation

3.3. Moderately Singular Equations

3.3.1. Problems

Chapter 4. The Laplace Transform

4.1. The Laplace Transform and Its Inverse

4.1.1. Review of Improper Integrals

4.1.2. The Laplace Transform

4.1.3. The Inverse Laplace Transform

4.2. Solving the Initial Value Problems

4.2.1. Step Functions

4.3. The Delta Function and Impulse Forces

4.4. Convolution and the Tautochrone Curve

4.4.1. The Tautochrone curve

4.5. Distributions

4.5.1. Problems

Chapter 5. Linear Systems ofDifferential Equations

5.1. The Case of Real Eigenvalues

5.1.1. Review of Vectors and Matrices

5.1.2. Linear FirstOrder Systems with Constant Coefficients

5.2. A Pair of Complex Conjugate Eigenvalues

5.2.1. ComplexValued and RealValued Solutions

5.2.2. The General Solution

5.2.3. Nonhomogeneous Systems

5.3. The Exponential of a Matrix

5.3.1. Problems

5.4. Floquet Theory and Massera’s Theorem

5.5. Solutions of Planar Systems Near the Origin

5.5.1. Linearization and the HartmanGrobman Theorem

5.5.2. Phase Plane and the Prüfer Transformation

5.5.3. Problems

5.6. Controllability and Observability

5.6.1. The CayleyHamilton Theorem

5.6.2. Controllability of Linear Systems

5.6.3. Observability

5.6.4. Problems

Chapter 6. Nonlinear Systems

6.1. The PredatorPrey Interaction

6.2. Competing Species

6.3. An Application to Epidemiology

6.4. Lyapunov’s Stability

6.4.1. Stable Systems

6.5. Limit Cycles

6.5.1. PoincaréBendixson Theorem

6.5.2. DulacBendixson Criterion

6.6. Periodic Population Models

6.6.1. Problems

Chapter 7. The Fourier Series and Boundary Value Problems

7.1. The Fourier Series for Functions of an Arbitrary Period

7.1.1. Even and Odd Functions

7.1.2. Further Examples and the Convergence Theorem

7.2. The Fourier Cosine and the Fourier SineSeries

7.3. TwoPoint Boundary Value Problems

7.3.1. Problems

7.4. The Heat Equation and the Method of Separation of Variables

7.5. Laplace’s Equation

7.6. The Wave Equation

7.6.1. Nonhomogeneous Problems

7.6.2. Problems

7.7. Calculating Earth’s Temperature and Queen Dido’s Problem

7.7.1. The Complex Form of the Fourier Series

7.7.2. The Temperatures Inside the Earth and Wine Cellars

7.7.3. The Isoperimetric Inequality

7.8. Laplace’s Equation on Circular Domains

7.9. SturmLiouville Problems

7.9.1. The FourierBessel Series

7.9.2. Cooling of a Cylindrical Tank

7.9.3. Cooling of a Rectangular Bar

7.10. Green’s Function

7.10.1. Problems

7.11. The Fourier Transform

7.12. Problems on Infinite Domains

7.12.1. Evaluation of Some Integrals

7.12.2. The Heat Equation for ∞<𝑥<∞

7.12.3. SteadyState Temperatures for the Upper HalfPlane

7.12.4. Using the Laplace Transform for a SemiInfinite String

7.12.5. Problems

Chapter 8. Elementary Theory of PDE

8.1. Wave Equation: Vibrations of an InfiniteString

8.2. SemiInfinite String: Reflection of Waves

8.3. Bounded String: Multiple Reflections

8.4. Neumann Boundary Conditions

8.5. Nonhomogeneous Wave Equation

8.5.1. Problems

8.6. FirstOrder Linear Equations

8.6.1. Problems

8.7. Laplace’s Equation: Poisson’s Integral Formula

8.8. Some Properties of Harmonic Functions

8.9. The Maximum Principle

8.10. The Maximum Principle for the Heat Equation

8.10.1. Uniqueness on an Infinite Interval

8.11. Dirichlet’s Principle

8.12. Classification Theory for Two Variables

8.12.1. Problems

Chapter 9. Numerical Computations

9.1. The Capabilities of Software Systems, Like Mathematica

9.2. Solving Boundary Value Problems

9.3. Solving Nonlinear Boundary Value Problems

9.4. Direction Fields

Appendix

A.1. The Chain Rule and Its Descendants

A.2. Partial Fractions

A.3. Eigenvalues and Eigenvectors

A.4. Matrix Functions and the Norm

Bibliography

Index

Back Cover

In this introduction to differential equations, the author emphasizes that he wanted to write a book that students would like to read. While theoretical results are mentioned, stated, and occasionally proved throughout the book, the focus is on solving differential equations and understanding properties of their solutions.
Bill Satzer, MAA Reviews