A First Course in Ordinary Differential Equations

✍ Scribed by Suman Kumar Tumuluri

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 338
• Edition: 1
• Category: Library

✦ Synopsis

A First course in Ordinary Differential Equations provides a detailed introduction to the subject focusing on analytical methods to solve ODEs and theoretical aspects of analyzing them when it is difficult/not possible to find their solutions explicitly. This two-fold treatment of the subject is quite handy not only for undergraduate students in mathematics but also for physicists, engineers who are interested in understanding how various methods to solve ODEs work. More than 300 end-of-chapter problems with varying difficulty are provided so that the reader can self examine their understanding of the topics covered in the text.

Most of the definitions and results used from subjects like real analysis, linear algebra are stated clearly in the book. This enables the book to be accessible to physics and engineering students also. Moreover, sufficient number of worked out examples are presented to illustrate every new technique introduced in this book. Moreover, the author elucidates the importance of various hypotheses in the results by providing counter examples.

Features

• Offers comprehensive coverage of all essential topics required for an introductory course in ODE.

• Emphasizes on both computation of solutions to ODEs as well as the theoretical concepts like well-posedness, comparison results, stability etc.

• Systematic presentation of insights of the nature of the solutions to linear/non-linear ODEs.

• Special attention on the study of asymptotic behavior of solutions to autonomous ODEs (both for scalar case and 2✕2 systems).

• Sufficient number of examples are provided wherever a notion is introduced.

• Contains a rich collection of problems.

This book serves as a text book for undergraduate students and a reference book for scientists and engineers. Broad coverage and clear presentation of the material indeed appeals to the readers.

Dr. Suman K. Tumuluri has been working in University of Hyderabad, India, for 11 years and at present he is an associate professor. His research interests include applications of partial differential equations in population dynamics and fluid dynamics.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
1. Introduction
1.1. Ordinary differential equations
1.2. Applications of ODEs
2. First order ODEs
2.1. A review of some basic methods
2.1.1. Separation of variables
2.1.2. Exact equations
2.1.3. Linear ODEs
2.2. Well-posedness
2.2.1. Continuable solutions
2.3. Differential inequalities
2.3.1. Applications of Gronwall's lemma
2.4. Comparison results
2.5. The first order scalar autonomous equations
3. Higher order linear ODEs
3.1. ODEs with constant coefficients
3.1.1. Factorization of di erential operators: homogeneous case
3.1.2. Factorization of di erential operators: non-homogeneous
3.1.2.1. Method of partial fractions
3.1.2.2. Power series method
3.1.2.3. Method of undetermined coe cients
3.1.2.4. Exponential shift rule
3.1.3. Euler's equation
3.2. ODEs with variable coefficients
3.2.1. Dimension of the solution space
3.2.2. Wronskian and its properties
3.2.3. Lagrange's method of reduction of the order
3.2.4. Zeros of the solutions to second order ODEs
3.3. Non-homogeneous ODEs with variable coefficients
3.3.1. Method of variation of parameters
4. Boundary value problems
4.1. Introduction
4.2. Adjoint forms
4.2.1. Boundary conditions
4.3. Green's function
4.3.1. Non-homogeneous boundary conditions
4.4. Sturm-Liouville systems and eigenvalue problems
5. Systems of first order ODEs
5.1. Introduction
5.2. Existence and uniqueness: Picard's method revisited
5.3. Systems of linear ODEs with constant coefficients
5.3.1. Exponential of a matrix and its properties
5.3.1.1. Working rule to nd eA
5.3.2. Solution to Y 0 = AY
5.4. Systems of linear ODEs with variable coefficients
5.4.1. Solution matrix and fundamental matrix
5.4.2. Non-homogeneous ODEs: method of variation of parameters revisited
6. Qualitative behavior of the solutions
6.1. Introduction
6.2. Linear systems with constant coefficients
6.3. Lyapunov energy function
6.4. Perturbed linear systems
6.5. Periodic solutions
7. Series solutions
7.1. Introduction
7.2. Existence of analytic solutions
7.3. The Legendre equation
7.3.1. Applications of Rodrigue's formula
7.4. Linear ODEs with regular singular points
7.5. Bessel's equation
7.6. Regular singular points at in nity
8. The Laplace transforms
8.1. Introduction
8.2. Definition and properties
8.2.1. The Heaviside function
8.2.2. The convolution
8.3. Inverse Laplace transforms
8.4. Applications to ODEs
9. Numerical Methods
9.1. Introduction
9.2. Euler methods
9.3. The Runge Kutta Method
Appendix A
A.1. Metric spaces
Appendix B
B.1. Another proof of the Cauchy-Lipschitz theorem
Appendix C
C.1. Some useful results from calculus
Bibliography
Index