Lecture
This lecture presents the construction of the general solution for differential equations with a second member, focusing on the superposition principle. The instructor introduces a theorem that outlines the structure of all solutions to these equations. The general solution is expressed as a combination of a homogeneous solution and a particular solution, with the latter being derived from the specific problem at hand. The lecture emphasizes the uniqueness of solutions to homogeneous equations, which allows for the derivation of all other solutions through the superposition of known solutions. The instructor provides a brief proof of the theorem, highlighting the importance of understanding the uniqueness of solutions. Practical applications are discussed, particularly in relation to solving exercises involving various types of second members, such as experimental and trigonometric functions. The lecture concludes with a preview of upcoming methods for calculating these solutions, reinforcing the linearity of the equations involved and the effectiveness of the proposed approach.