The main objective of this work is to teach students to analyse particular solutions of initial value problems for homogeneous and inhomogeneous differential equations of the second order  with a constant coefficient from the theory of ordinary differential equations. The work contains a list of exercises considered by the students in independent work with the teacher. A solution and analysis of the list of exercises discussed is given. A list of tasks offered for students' independent work on the topic under consideration is presented. It was also concluded about the performance of the students' assignments. Student's activity in independent works increases student's interest in conducting scientific research. It is known in scientific research that problem-solving analysis and forecasting are highly valued abilities. We believe that the knowledge and experience gained by students through independent work will have a significant impact on their professional development.