A First Course in Ordinary Differential Equations

Suman Kumar Tumuluri

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000356755
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

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

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

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

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

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

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

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• 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.

Chapter 1

Introduction

1.1 Ordinary differential equations

The term ‘equatio differentialis’ (differential equations) was first used by Leibniz in 1676 to denote a relationship between the differentials of two variables. Very soon, this restricted usage was abandoned. Roughly speaking, differential equations are the equations involving one or more dependent variables (unknowns) and their derivatives/partial derivatives. If the unknown in the differential equation is a function of only one variable, then such differential equation is called an ordinary differential equation (ODE).

Notation: Unless specified otherwise, the unknown in the differential equation is denoted by y. Let ℝ denote the set of real numbers, and J be an open interval in ℝ. Throughout the book we denote the derivative of the function $y:J\to ℝ$ with respect to x by either

$$\begin{array}{l}\frac{d}{dx}y(x)\text{or}\frac{dy}{dx}(x)\text{or}y\prime (x).\hfill \end{array}$$

When there is no ambiguity regarding the argument in the function y, we denote the derivative simply with $\frac{dy}{dx}$ or y′. Similarly, let y″ and y^″′ denote the second and the third derivative of y, respectively. In general, for $k\in \mathbb{N},$ $y^{(k)}$ or $\frac{d^{k}y}{d{x}^k}$ denotes the k-th order derivative of y.

With this notation, examples of ODEs are

$$\begin{array}{l}\frac{d}{dx}y(x)={(\frac{d^2}{d{x}^2}y(x))}^5+y^2(x),x\in (0,1),\hfill \end{array}$$(1.1)

$$\begin{array}{l}y\prime =3y^2+(\sin x)y+\log ({\cos }^{2}y),x\in ℝ.\hfill \end{array}$$(1.2)

The order of an ODE is the largest number k such that the k-th order derivative of the unknown is present in the ODE. For example, the order of (1.1) is two.

At the beginning, it may look like tools from the integral calculus are sufficient to study ODEs. But very soon one realizes that to develop methods to solve or analyze them, one needs notions from subjects like analysis, linear algebra, etc. In fact, the study of differential equations motivated crucial development of many areas of mathematics: the theory of Fourier series and more general orthogonal expansions, integral transformations, Hilbert spaces, and Lebesgue integration to name a few.

1.2 Applications of ODEs

Many laws in physics, chemistry, biology etc., can be easily expressed using differential equations. One of the reasons for this is the following. The quantity $y\prime (x)$ can be interpreted as the rate of change of the quantity y with respect to the quantity x. In many natural phenomena, there is a relationship between the unknowns (which are relatively difficult to measure), the rate of change of the unknowns with respect to a known quantity, and the other known quantities (which are easy to measure) that govern the process. When this relationship is expressed in mathematics, it turns out to be a (system of) differential equation(s). Therefore th...