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| 1 | The paper presents the results of involvement of future mathematics and physics teachers in independent scientific research activity in conditions of realization of competence approach. In the case of such approach the competence is a general capability which is checked and created in activities. It is based on knowledge and allows the person to establish connection between the system of actions for successful problem solution. During the educational process, the formation of elements of competencies at implementation of curriculum is carried out by means of maintenance of the studied subject. The course of the differential equations has great opportunities for forming of professional competencies of future teachers of mathematics and physics, however the existing education guidance, according to the theory of differential equations, are obviously not oriented to form the competencies. Besides modern development of technology, chemistry, biology, ecology, geography, economics and other sciences is impossible without the use of differential equations. The solutions of Clairaut-type equations with a special right-hand part found in the paper are a new result in the theory of partial differential equations. In turn the organization of independent scientific research in the framework of the course of differential equations promotes the main goal of professional competencies of future mathematics and physics teachers, namely forming the professional competencies being ready to use theoretical and practical knowledge in science and education. Keywords: differential equations, professional competencies | 1563 | ||||