Differential Equations for Engineers and Scientists vol.2

This book is the second volume of my book series on Differential Equations. All definitions, terminology, fundamental concepts and techniques for solving first order differential equations are presented in full details and considerable depth in my textbook “Differential Equations for Engineers and Scientists, Vol. 1”. This volume also contains a considerable number of applications in Engineering, Geometry, Physics, etc. The current second volume is actually a continuation of the first volume. References to Vol. 1 are frequently made, either by the chapter number and/or the equation number. It becomes thus obvious that the first volume is a prerequisite for the understanding of the material developed in this volume.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.In this textbook we lay out fundamental concepts and definitions, investigate various types of second and higher order differential equations, present methods of obtaining their general solution and finally show how to solve a variety of problems arising in Engineering and Physics by means of their corresponding differential equations.The book contains 17 chapters, as shown analytically in the Table of Contents. The first three chapters are devoted to fundamental concepts, definitions and terminology of second order differential equations, (differential equation of a two parameters family of curves, initial and boundary value problems, second order equation with the independent or the dependent variable missing, etc). Chapters 4 up to 11 are devoted exclusively to the study of second order, linear differential equations, with constant and variable coefficients, (linearly independent functions, the determinant of Wronski, Cauchy’s theorem about the existence and the uniqueness of the solution, complementary, particular and general solutions, Lagrange’s method for the particular solution, equations with variable coefficients reducible to ones with constant coefficients, Euler’s equation, Bessel’s and Legendre’s equation, etc). The general theory of linear differential equations of higher order is developed in chapters 12, 13 and 14. Operator methods of solving differential equations are presented in chapter 15, while systems of differential equations are considered in chapter 16. Finally chapter 17 is devoted to the solution of a variety of problems in Engineering and Physics, (free and forced mechanical vibrations with and without damping, free and forced electrical oscillations, series and parallel R-L-C circuits, transients in mechanical and electrical systems and steady state response to sinusoidal excitation by means of the complex phasor’s method, power and energy in circuits, the impulse response of second order systems, etc).The book includes 115 illustrative worked out examples and 220 graded problems for solution. 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.