This chapter provides some basic material on linear ordinary differential equations of the first and second order. The concepts of ordinary differential equation, order of a differential equation, linearity, solution of an ordinary differential equation, Wronskian, and the Cauchy problem are presented. Also, some special first-order nonlinear ordinary differential equations are mentioned, namely the Bernoulli and Riccati equations. We then turn our attention to the linear first-order ordinary differential equation, for which we discuss a few elementary integration methods. We also mention two specific nonlinear first-order differential equations, the Bernoulli and Riccati equations. For the second-order ordinary differential equations, particularly, equations with constant coefficients and equations of the Euler type are discussed. At the end, several solved exercises are discussed step by step, concluding with a list of proposed exercises, among which some interesting applications are left to the reader—Legendre equation, harmonic oscillator, Whittaker equation, RLC-circuit, and the Green function—all of them with the corresponding anwser and/or suggestion.