  The development of functions of real variables in Taylor and Frobenius series
(whole series which are formed in nonorthogonal, nonperiodic bases), in
sinusoidal Fourier series (bases of orthogonal, periodic functions), in series
of special functions (bases of orthogonal, nonperiodic functions), etc. is a
commonly used method for solving a wide range of ordinary differential
equations (ODEs) and partial differential equations (PDEs).In this article,
based on an in-depth analysis of the properties of periodic sinusoidal Fourier
series (SFS), we will be able to apply this procedure to a much broader
category of ODEs (all linear, homogeneous and non-homogeneous equations with
constant coefficients, a large category of linear and non-linear equations with
variable coefficients, systems of ODEs, integro-differential equations, etc.).
We will also extend this procedure and we use it to solve certain ODEs, on
non-orthogonal periodic bases, represented by non sinusoidal periodic Fourier
series (SFN).
