We propose a modification of a method based on Fourier analysis to
obtain the Floquet characteristic exponents for periodic homogeneous
linear systems, which shows a high precision. This modification uses a
variational principle to find the correct Floquet exponents among the
solutions of an algebraic equation. Once we have these Floquet
exponents, we determine explicit approximated solutions. We test our
results on systems for which exact solutions are known to verify the
accuracy of our method including one dimensional periodic potentials of
interest in quantum physics. Using the equivalent linear system, we also
study approximate solutions for homogeneous linear equations with
periodic coefficients.