The use of integral transforms to obtain solutions of linear differential equations consists of transforming a given linear differential equation into a simpler equation, solving this new equation, and finally, calculating the corresponding inverse transform to obtain the solution of the initial linear differential equation. This method of transforms is used to obtain particular solutions of linear differential equations, ordinary and partial. We discuss here only two such transforms, the Laplace transform and the Fourier transform. We present some properties of Laplace and Fourier transforms involving their derivatives and convolutions, as well as the ways to obtain the corresponding inverse Laplace and Fourier transforms using integration on the complex plane. Some solved exercises are discussed step by step, specifically in the study of a linear ordinary differential equation and inversion problem involving the modified Bromwich contour. As an application we evaluate the Fourier transform of the Dirac delta function and obtain an integral representation for the Bessel function. We conclude with a list of proposed exercises; among them, some interesting applications involving Green’s function are left to the reader, all of them with the answer and/or suggestion.