Solutions of linear ordinary differential equations with non-singular varying coefficients by using the corrected Fourier series

Solutions of linear ordinary differential equations (ODEs) with non-singular varying coefficients are constructed by using the corrected Fourier series [Q.H. Zhang, S. Chen, Y. Qu, Corrected Fourier series and its application to function approximation, Int. J. Math. Math. Sci. (1) (2005) 33-42]. In essence our method is a Galerkin method with the corrected Fourier series as its basis functions. For mth order ODEs the m linearly independent solutions are uniformly convergent until their mth derivatives, i.e., no Gibbs oscillations in the solutions themselves and in their derivatives until mth order over the ODE's entire interval. Procedures of obtaining two (three) linearly independent Galerkin solutions are presented for second (third) order ODEs.