LINEAR DIFFERENTIAL EQUATIONS AND OSCILLATORS (HB 2020) Hardcover

Linear Differential Equations and Oscillators is the first book in the six-volume series Ordinary Differential Equations with Applications to Trajectories and Vibrations, which is part of the Mathematics and Physics Applied to Science and Technology collection. This book contains the first two chapters of the fourth volume. The first chapter focuses on linear differential equations of any order, whose unforced solutions can be derived from the roots of the characteristic polynomial. These equations include: (i) those with constant coefficients, (ii) those with homogeneous power coefficients where the exponent equals the order of derivation, and (iii) linear finite difference equations with constant coefficients. The chapter discusses both unforced and forced solutions, providing general insights into the properties of ordinary differential equations. The second chapter applies the principles from the first chapter to linear second-order oscillators with a single degree of freedom, such as mechanical mass-damper-spring systems and electrical resistor-capacitor circuits. It covers free undamped, damped, and amplified oscillations, as well as forced oscillations, including beats, resonance, discrete and continuous spectra, and impulsive inputs. The book explores the general properties of differential and finite difference equations, emphasizing linear equations with constant and some power coefficients, and presents both particular and general solutions for various cases. It also addresses solutions to different forcing scenarios, including resonant cases, and discusses methods such as the characteristic polynomial, Green's functions, trigonometric series, Fourier integrals, and Laplace transforms.