Lecture
This lecture presents a detailed example of solving a Cauchy problem involving a differential equation. The instructor begins by introducing the equation u'(t) + u(t) = t² + sin(t) - e^(-t) with the initial condition u(0) = u₀. The lecture progresses through the steps of solving the equation, starting with the homogeneous part, where the solution is derived as an exponential function. The instructor then addresses the particular solution, breaking it down into components: a polynomial of degree two, a trigonometric function, and an exponential function. Each component is analyzed separately, with the instructor demonstrating how to derive coefficients by equating terms. The final solution combines all parts, and the instructor concludes by integrating the initial condition to find the complete solution. This structured approach provides a comprehensive understanding of solving differential equations with specific initial conditions.