Besides the elementary functions usually studied in basic calculus courses, there exists another (huge) class of functions called special functions, the knowledge of which is essential for solving many real problems arising in exact sciences. These functions appear almost always as sets of solutions of some particular family of ordinary differential equations. We investigate the solutions of a general linear second-order ordinary differential equation with three regular singular points, which will be transformed into a hypergeometric equation, whose solutions are the hypergeometric functions. By considering a certain limit process applied to the hypergeometric equation, we will obtain the confluent hypergeometric equation, whose solutions are the confluent hypergeometric functions. Some solved exercises are discussed step by step, specifically involving the hypergeometric functions and their particular cases. As an application we discuss the so-called projective Laplace equation written in spherical coordinates and the generating function for the Bessel functions. We conclude with a list of proposed exercises; some interesting applications are left to the reader, particularly ones involving Laguerre, Hermite, and Weber functions, all of them with the anwser and/or suggestion.