The world abounds with introductory texts on ordinary differential equations and rightly so in view of the large number of students taking a course in this subject. However, for some time now there is a growing need for a junior-senior level book on the more advanced topics of differential equations. In fact the number of engineering and science students requiring a second course in these topics has been increasing. This book is an outgrowth of such courses taught by us in the last ten years at Worcester Polytechnic Institute. The book attempts to blend mathematical theory with nontrivial applications from varipus disciplines. It does not contain lengthy proofs of mathemati~al theorems as this would be inappropriate for its intended audience. Nevertheless, in each case we motivated these theorems and their practical use through examples and in some cases an "intuitive proof" is included. In view of this approach the book could be used also by aspiring mathematicians who wish to obtain an overview of the more advanced aspects of differential equations and an insight into some of its applications. We have included a wide range of topics in order to afford the instructor the flexibility in designing such a course according to the needs of the students. Therefore, this book contains more than enough material for a one semester course.

Inhalt

O: Review.- 1. Solution of second order ordinary differential equations by series.- 2. Regular singular points.- 3. Series solutions near a regular singular point.- 1: Boundary Value Problems.- 1. Introduction.- 2. Adjoint differential equations and boundary conditions.- 3. Self -adjoint systems.- 4. A broader approach to self-adjoint systems.- 5. Sturm-Liouvi1 le theory.- 6. Introduction to orthogonality and completeness.- 2: Special Functions.- 1. Hypergeometric series.- 2. Bessel functions.- 3. Legendre polynomials.- 4. Gamma function.- 3: Systems of Ordinary Differential Equations.- 1. Introduction.- 2. Method of elimination.- 3. Some linear algebra.- 4. Linear systems with constant coefficients.- 5. Linear systems with variable coefficients.- 6. Elements of linear control theory.- 7. The Laplace transform.- 4: Applications of Symmetry Principles to Differential Equations.- 1. Introduction.- 2. Lie groups.- 3. Lie algebras.- 4. Prolongation of the action.- 5. Invariant differential equations.- 6. The factor ization method.- 7. Examples of factorizable equations.- 5: Equations with Periodic Coefficients.- 1. Introduction.- 2. Floquet theory for periodic equations.- 3. Hill's and Mathieu equations.- 6: Green's Functions.- 1. Introduction.- 2. General definition of Green's function.- 3. The interpretation of Green's functions.- 4. Generalized functions.- 5. Elementary solutions and Green's functions.- 6. Eigenfunetion representation of Green's functions.- 7. Integral equations.- 7: Perturbation Theory.- 1. Preliminaries.- 2. Some basic ideas-regular perturbations.- 3. Singular perturbations.- 4. Boundary layers.- 5. Other perturbation methods.- 6. Perturbations and partial differential equations.- 7. Perturbation of eigenvalue problems.- *8. The Zeeman and Stark effects.- 8: Phase Diagrams and Stability.- 1. General introduction.- 2. Systems of two equations.- 3. Some general theory.- 4. Almost linear systems.- 5. Almost linear systems in R2.- 6. Liapounov direct method.- 7. Periodic solutions (limit cycles).- 9: Catastrophes and Bifurcations.- 1. Catastrophes and structural stability.- 2. Classification of catastrophe sets.- 3. Some examples of bifurcations.- 4. Bifurcation of equilibrium states in one dimension.- 5. Hopf bifurcation.- 6. Bifurcations in R.- 10: Sturmian Theory.- 1. Some mathematical preliminaries.- 2. Sturmian theory for first order equations.- 3. Sturmian theory for second order equations.- 4. Prufer transformations.