Lecture
This lecture focuses on solving a Cauchy problem using the method of variation of constants. The instructor begins by presenting the differential equation u'(t) + t u(t) = 2t, with the initial condition u(0) = U0. The lecture revisits the general solution approach discussed in previous sessions, emphasizing the construction of the solution without relying solely on the formula. The instructor explains how to derive the solution by varying the constant C, transforming it into a function of t. This method allows for a more intuitive understanding of the problem. The lecture details the steps to derive the expression for u(t) by integrating and applying the initial condition. The final solution incorporates both the particular and homogeneous parts of the equation, demonstrating the relationship between the general solution and the specific initial condition. The instructor concludes by highlighting the importance of understanding the construction of solutions in differential equations, particularly in the context of Cauchy problems.