The Linear Equivalence Method - briefly, LEM - is author's original method. She created it to the purpose of determining and studying the solutions of nonlinear dynamical systems depending on parameters in a classical linear frame and firstly published the idea of LEM 1978. She published two books on LEM in Romanian. The present book is a revised and completed edition of the last one, appeared in 2008 at Editura Tehnica and prefaced by prof. P.P.Teodorescu; it includes, besides the previous results, many other theoretical considerations and applications in a multitude of domains, realized by her and/or by other authors. It is well known that the nonlinear problems involve much more difficulties than the linear ones. Basically, the study of these problems begins with their linearization, which often means to cut off the nonlinear terms or to approximate them by linear ones, which is far from being satisfactory. LEM overcomes this difficulty, establishing a one-to-one correspondence between the solutions of an initial polynomial problem and the analytic solutions of a certain linear PDE. This justifies the denomination of LEM - the Linear Equivalence Method. LEM was extended to more general cases of nonlinear ODSs. There are two important LEM advantages: the local inverse of a nonlinear operator, which gives rise to specific LEM convergent algorithms and the normal LEM representation, useful in qualitative studies of solutions. LEM is theoretically founded in the first three chapters; in the fourth, there are realized connections of LEM with other domains, such as Fliess series, Lie derivatives, theory of graphs, theory of numbers, etc. LEM proves itself a very efficient tool for the study of a great variety of problems: the nonlinear deformation of straight bars or the motion of relativistic charged particles, the buckling of a straight bar, the ecological prey-predator model leading to the Lotka-Volterra system, the Belousov-Zhabotinskij oscillatory chemical reaction, etc. Other authors successfully applied LEM to their nonlinear models, e.g. the nonlinear double pendulum, heavy elastica, neo-Hookean or Ogden type models. These applications are presented in the last four chapters. One should mention the pertinent applications of LEM presented in chapters 5 and 6, realized by the author in co-operation with the regretted prof. dr. rer. nat. Petre P.Teodorescu. He was a professor of Mechanics at the faculty of Mathematics of the University of Bucharest, the head of the department of Mechanics of the Academy of Technical Sciences of Romania and a well-known author of numerous important papers and treatises concerning the mechanics of solids. Professor P.P.Teodorescu's testimonial to LEM is (from the preface of the book on LEM appeared in 2008): "The method proposed by Ileana Toma is remarkable; having the privilege of applying it, I am aware that LEM represents a powerful tool, convenient for the study of nonlinear problems, the more so useful as it can be applied to a wide range of problems. The appearance of an original scientific book always represents a special event; but I consider this particular book exceptional, as, along with the purely mathematical contribution, it also contains numerous new valuable results in various domains of the natural sciences." This book is addressed to all the specialists in mathematics and/or in other domains as mechanics, engineering, physics, chemistry, biology, a.s.o., who deal with models of the form of nonlinear ordinary differential equations. The understanding of the method requires only elementary notions of differential equations, analysis and algebra: LEM was applied also by students in their graduation projects. In fact, its purpose is to provide the users with an efficient tool that maps one to one the nonlinear to the linear frame. This enables the free application of the methods of the linear, much easier handled and better founded that those of the nonlinear.