Tobias Holck Colding continues MIT's 18.100B Real Analysis course with a lecture on power series and the start of ordinary differential equations. He proves that a power series can be differentiated and integrated term by term within its radius of convergence, working through the justification rather than just stating the result. The second half turns to ODEs, introducing what they are and laying out the conceptual framework for existence and uniqueness of solutions, the question of whether a differential equation actually has a solution and whether that solution is the only one. The lecture runs eighty one minutes in the standard chalkboard format of MIT OpenCourseWare recordings, building directly on prior material in the course sequence and aimed at students already comfortable with convergence and series fundamentals.