Differential Equations for Engineers and Scientists vol.1

Differential equations play an extremely important role in almost all branches of Engineering, Mathematics, Physics, etc. and in particular in the study of Applied Mathematics.Many Physical and Engineering problems, when formulated in mathematical terms, lead to differential equations. The motion of a simple pendulum, the orbit of planets, the vibrations of a mechanical system and the oscillations of an electrical system, just to mention a few are described in terms of differential equations.In general terms, one dimension problems are described by Ordinary Differential Equations (O.D.E) while higher dimension problems are described by Partial Differential Equations (P.D.E). The Schrodinger’s equation in Quantum Mechanics and the Einstein’s Field equations in General Relativity are Partial Differential Equations. It seems that our world is governed by differential equations.The subject of differential equations is a complex one. In this text we lay out fundamental concepts and definitions, investigate various types of first order differential equations, present methods of obtaining their general solution and finally show how to solve a variety of problems arising in Engineering, Physics, Geometry, Biology, and Economy by means of the corresponding differential equations.This book contains 25 chapters, as shown analytically in the Table of Contents. The first six chapters are devoted to fundamental concepts, definitions and terminology on differential equations. In particular we show how to isolate the solution of an equation subject to an additional initial or boundary condition (initial value or boundary value problems). In the next twelve chapters we investigate the main types of first order differential equations and develop methods to obtain their general solution, (separable variables, homogeneous equations, linear equations, Bernoulli’s equation, exact equations and equations reducible to exact, integrating factors, Clairaut’s equation, Lagrange’s equation, integration by means of differentiation, interchange of x and y, the general method of substitution). The important Cauchy’s theorem on the existence and the uniqueness of the solution of a differential equation is stated in chapter 19 while a preliminary introduction on the approximate (numerical) methods of solution is given in chapter 20. Chapters 21, 22, 23 and 24 are exclusively devoted to a great variety of applications arising in Physics, Geometry, Engineering, Economy, Biology, etc.Finally, in the last chapter we give a brief introduction of the extremely useful “Dirac’s Delta function δ(t)” or “impulse function” (introduced by Paul Dirac in his study of Quantum Mechanics) and investigate various properties and interesting applications. The book includes 130 illustrative worked out examples and 285 graded problems to be solved. The examples and the problems are designed to help students to develop a solid background in differential equations and applications, to broaden their knowledge and sharpen their analytical skills and finally to prepare them to pursue successfully more advanced studies in Engineering and Mathematics.The reader of this book should have a good working knowledge of ordinary and partial differentiation and differential and integral calculus in general.A second volume on differential equations of second and higher order (Theory and Applications) will be released in the near future.