Abstract
Here, we propose a method to obtain local analytic approximate solutions
of ordinary differential equations with variable coefficients, or even
some non-linear equations, inspired in the Lyapunov method, where
instead of polynomial approximations, we use truncated Fourier series
with variable coefficients as approximate solutions. In the case of
equations admitting periodic solutions, an averaging over the
coefficients gives global solutions. We show that, under some
restrictive condition, the method is equivalent to the
Picard-Lindel\”of method. After some numerical
experiments showing the efficiency of the method, we apply it to
equations of interest in Physics, in which we show that our method
possesses an excellent precision even with low iterations.