This book outlines the basics of the theory of ordinary differential equations, partial first-order equations, and the basis of variational calculus. It is written on the basis of a course of lectures, which the author read at the Moscow Institute of Physics and Technology for more than ten years. The book is designed for students of higher technical educational institutions (with a course of higher mathematics of 510 hours or more) and for research engineers. The book most fully presents the sections of the theory of ordinary differential equations related to the problems of small oscillations and propagation of linear waves. The last chapter is devoted to one of the most important asymptotic methods - the VKB method for ordinary linear differential equations of the second order .. The book includes some concepts of functional analysis and a number of necessary information from mathematical analysis. For students of technical universities and research engineers .