Lecture
This lecture focuses on solving a Cauchy problem using the general form of solution construction. The instructor presents a specific differential equation involving a first-order linear equation. The equation is analyzed, and the instructor explains how to derive the general solution by applying a formula that includes a constant multiplied by an exponential function and the primitive of a given function. The lecture emphasizes the importance of finding a particular solution and constructing the exponential term. The instructor demonstrates the calculation of the primitive and how it relates to the initial condition of the problem. By substituting the initial value into the derived general solution, the instructor determines the constant that satisfies the initial condition. The final result is presented as the unique solution defined over the interval from zero to infinity, highlighting the solution's global and unique nature. This structured approach to solving the Cauchy problem provides a clear understanding of the methods involved in differential equations.