Linear Differential Equations and Oscillators is the first book withinOrdinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set.As a set, they are the fourth volume in the seriesMathematics and Physics Applied to Science and Technology. This first book consists of chapters 1 and 2 of the fourth volume.The first chapter covers linear differential equations of any order whose unforced solution canbe obtained from the roots of a characteristic polynomial, namely those: (i) with constantcoefficients; (ii) with homogeneous power coefficients with the exponent equal to the order ofderivation. The method of characteristic polynomials is also applied to (iii) linear finite differenceequations of any order with constant coefficients. The unforced and forced solutions of (i,ii,iii) areexamples of some general properties of ordinary differential equations.The second chapter applies the theory of the first chapter to linear second-order oscillators withone degree-of-freedom, such as the mechanical mass-damper-spring-force system and theelectrical self-resistor-capacitor-battery circuit. In both cases are treated free undamped, damped,and amplified oscillations; also forced oscillations including beats, resonance, discrete andcontinuous spectra, and impulsive inputs.Describes general properties of differential and finite difference equations, with focus on linear equations and constant and some power coefficientsPresents particular and general solutions for all cases of differential and finite difference equationsProvides complete solutions for many cases of forcing including resonant casesDiscusses applications to linear second-order mechanical and electrical oscillators with dampingProvides solutions with forcing including resonance using the characteristic polynomial, Green' s functions, trigonometrical series, Fourier integrals and Laplace transforms