The dissertation paper deals with teaching mathematics to students of technical higher educational institutions. Having studied the problem we are in position to state that educational and cognitive activities of a student are stimulated by using systematically information and communication technologies. This being the case, a methodological support for studying differential equations was developed. It comprises a handbook providing an intradiscipline approach to studying differential equations, an exercise book in differential equations for stimulating individual work of a student, a teaching aid aimed at increasing the applied orientation in teaching differential equations, a distance course in differential equations for organizing the blended learning based educational process which allows also organizing the educational process in advance, using the distance course contents as a material for student individual work and pre-practicum training, providing interactive an self-control. The dissertation paper presents a scientifically grounded and experimentally verified methodology of teaching differential equations to prospective mechanical engineers. Developing the methodology involved initiating the conceptual framework; creating objective models of mathematical abilities of a student and competencies of a lecture of a technical higher educational institution; developing a blended learning based methodology of teaching differential equations aimed at combining traditional and innovational methods with information and communication technologies; developing appropriate implementation tools such as criteria, levels, methods, forms, and means; determining the efficiency of the proposed methodology; developing a model for implementing the methodology based on combination of individual-differentiated, personal-active, and competence approaches; characterizing its structural components such as target, content, organization and technology, control, and evaluation. In the dissertation we specify the term "blended learning" as a way of organizing educational process providing integration of student and teacher in-class and out-of-class work, traditional educational and information and communication technologies. We determine a mechanical engineer mathematical competence structure containing a list of components necessary for successful study of differential equations according to the requirements of educational qualification characteristics for a graduate of respective specialty. We improve the program of the differential equations contents modulus in the course of higher mathematics for competence based higher educational institution engineering students, contents and methodology for lectures, practices, individual work of a student according to the blended learning model. The forms and methods (slide lectures, seminars, professionally oriented creative projects, etc.) and means (a system of split-level computer supported training problems, presentations, tests, a distance course, forums) of the methodology of teaching differential equations to prospective mechanical engineers are further developed. The experimental verification of the dissertation statements shows that introducing the developed methodology of teaching differential equations to prospective mechanical engineers in the educational process contributes to increasing its intensity and efficiency, motivation settings, residual knowledge and skills quality indicators, improving the level of adequate competencies in differential equations, as well as stimulating the educational and cognitive activity and engineering thinking of a student. Thus the experimental results confirm the efficiency of the methodology developed. Based on the results of the experiment the following improvements of the educational process in the experimental groups compared to that in the control groups can be pointed out: a more deliberate mastering of the essence of educational material; increasing the student adequate competencies level; the students are positively motivated when studying differential equations; the students are interested in studying differential equations and applying them to solving engineering problems; improving student computer programming skills; improving student performance; the students are seeking self-fulfillment and self-control. The theoretical statements and methods of teaching differential equations to students of engineering departments of technical higher educational institutions using blended learning approach presented in the dissertation paper can be applied when developing special courses in mathematics aimed at improving professional skills of prospective mechanical engineers.
