Several solved exercises are discussed step-by-step, some of them as applications, for instance: vertical launch of an object, fall of a body with air resistance, Taylor series and the Frobenius method, Laurent series and residues, special functions, Hermite polynomials, Fourier-Bessel series and Fourier-Legendre series, Laplace transform and Fourier transform, Sturm-Liouville systems, Legendre polynomials, generalized Green’s function, linear partial differential equations, separation of variables, Laplace equation in spherical coordinates, Poisson equation, Helmholtz equation in cylindrical-parabolic coordinates, wave equation, left/right-sided Riemann-Liouville fractional integrals, memory effect and fractional order viscoelastic object. We conclude with a short list of proposed exercises, among which some interesting applications left to the reader, such as a fractional Maxwell model of elasticity and the Volterra integral equation, all of them with the corresponding answer and/or suggestion.