Ordinary and partial differential equations for the beginner (Q2813030)

As it is suggested by its title, this textbook contains an introduction to the theory of ordinary and partial differential equations. Its first notable feature is the distribution of the material in short topics (most of them 3--4 pages long) followed by problem sections. This might be quite convenient for a lecturer trying to organize his/her notes for a course in differential equations. Answers to selected problems are given at the end of the book.NEWLINENEWLINEThe emphasis is mainly put on the ordinary differential equations, but as it is pointed out in the preface, the book also includes those parts of the theory of first and second order partial differential equations, where the theory and solution methods are closely related to ordinary differential equations. The presentation is concise and in general self-contained.NEWLINENEWLINEThe first four chapters are devoted to the theory of first order ordinary differential equations. Chapter 1 introduces the basic terminology and the existence and uniqueness theory. In addition, differential equations with parameters are considered. Chapter 2 includes a presentation of the elementary solution methods. The first order linear ordinary differential equations in finite dimensional vector spaces are briefly considered in Chapter 3. It should be noted that the definition of exponential matrix and its main properties are not given in the text, so a reader who is unfamiliar with it has to find it in other source. Yet, a practical guidance for the computation of the exponential matrix is given. Chapter 4 is devoted to the concept of functional dependence/independence. The main goal is the theorem for the existence of a full system of functionally independent first integrals.NEWLINENEWLINEChapter 5 contains several topics related to higher order differential equations. First, the equivalent transition to a first order system of equations is given. Next, the technique of order reduction with intermediate integrals is presented. Then the focus moves to topics concerning higher order linear differential equations -- order reduction using the Wronskian and Liouville theorem, Euler equations, equations with constant coefficients and right side of the form of exponential polynomial, second order boundary value problems, Green function and power series solution method. Laplace transform is briefly described as well. Special attention is paid to Fourier transform for exponential polynomials. First order partial differential equations are introduced in Chapter 6 starting with homogeneous linear equations and then extending to the quasilinear case. The theory of characteristics for general first order equations is presented in Chapter 7. The discussion continues with higher order partial differential equations in Chapter 8 where the theorems of Kovalevskaya and Holmgren are given without proofs. A method for finding exponential polynomial solutions to evolution type linear partial differential equations is presented as well. The classification and normal forms for second order linear partial differential equations are discussed in Chapter 9. The final Chapter 10 is devoted to several special problems in two variables for hyperbolic equations -- Goursat, Cauchy and mixed problems, as well as, the Fourier method.