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Lectures on Differential Equations
 
Philip L. Korman University of Cincinnati, Cincinnati, OH
Lectures on Differential Equations
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-4704-5173-8
Product Code:  TEXT/54
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-5364-0
Product Code:  TEXT/54.E
List Price: $79.00
MAA Member Price: $59.25
AMS Member Price: $59.25
Hardcover ISBN:  978-1-4704-5173-8
eBook: ISBN:  978-1-4704-5364-0
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
Lectures on Differential Equations
Click above image for expanded view
Lectures on Differential Equations
Philip L. Korman University of Cincinnati, Cincinnati, OH
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN:  978-1-4704-5173-8
Product Code:  TEXT/54
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-5364-0
Product Code:  TEXT/54.E
List Price: $79.00
MAA Member Price: $59.25
AMS Member Price: $59.25
Hardcover ISBN:  978-1-4704-5173-8
eBook ISBN:  978-1-4704-5364-0
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 Details
     
     
    AMS/MAA Textbooks
    Volume: 542019; 399 pp
    MSC: 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 year-long 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 well-developed 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:

    Readership

    Undergraduate and graduate students interested in differential equations.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Copyright
    • Contents
    • Introduction
    • Chapter 1. First-Order Equations
    • 1.1. Integration by Guess-and-Check
    • 1.2. First-Order 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. Second-Order Equations
    • 2.1. Special Second-Order 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 Second-Order 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 Second-Order 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 First-Order Systems with Constant Coefficients
    • 5.2. A Pair of Complex Conjugate Eigenvalues
    • 5.2.1. Complex-Valued and Real-Valued 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 Hartman-Grobman Theorem
    • 5.5.2. Phase Plane and the Prüfer Transformation
    • 5.5.3. Problems
    • 5.6. Controllability and Observability
    • 5.6.1. The Cayley-Hamilton Theorem
    • 5.6.2. Controllability of Linear Systems
    • 5.6.3. Observability
    • 5.6.4. Problems
    • Chapter 6. Nonlinear Systems
    • 6.1. The Predator-Prey 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. Poincaré-Bendixson Theorem
    • 6.5.2. Dulac-Bendixson 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. Two-Point 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. Sturm-Liouville Problems
    • 7.9.1. The Fourier-Bessel 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. Steady-State Temperatures for the Upper Half-Plane
    • 7.12.4. Using the Laplace Transform for a Semi-Infinite String
    • 7.12.5. Problems
    • Chapter 8. Elementary Theory of PDE
    • 8.1. Wave Equation: Vibrations of an InfiniteString
    • 8.2. Semi-Infinite 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. First-Order 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 542019; 399 pp
MSC: 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 year-long 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 well-developed 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:

Readership

Undergraduate and graduate students interested in differential equations.

  • Cover
  • Title page
  • Copyright
  • Contents
  • Introduction
  • Chapter 1. First-Order Equations
  • 1.1. Integration by Guess-and-Check
  • 1.2. First-Order 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. Second-Order Equations
  • 2.1. Special Second-Order 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 Second-Order 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 Second-Order 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 First-Order Systems with Constant Coefficients
  • 5.2. A Pair of Complex Conjugate Eigenvalues
  • 5.2.1. Complex-Valued and Real-Valued 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 Hartman-Grobman Theorem
  • 5.5.2. Phase Plane and the Prüfer Transformation
  • 5.5.3. Problems
  • 5.6. Controllability and Observability
  • 5.6.1. The Cayley-Hamilton Theorem
  • 5.6.2. Controllability of Linear Systems
  • 5.6.3. Observability
  • 5.6.4. Problems
  • Chapter 6. Nonlinear Systems
  • 6.1. The Predator-Prey 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. Poincaré-Bendixson Theorem
  • 6.5.2. Dulac-Bendixson 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. Two-Point 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. Sturm-Liouville Problems
  • 7.9.1. The Fourier-Bessel 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. Steady-State Temperatures for the Upper Half-Plane
  • 7.12.4. Using the Laplace Transform for a Semi-Infinite String
  • 7.12.5. Problems
  • Chapter 8. Elementary Theory of PDE
  • 8.1. Wave Equation: Vibrations of an InfiniteString
  • 8.2. Semi-Infinite 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. First-Order 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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