In this paper we examine the linear differential equations x′ = Ax + P(φ)x, φ′ = ω where x ∈ Rn, φ ∈ Rm, A and ω are constant and P(φ) is real analytic and periodic in φ. We use the method of accelerated convergence to overcome the small divisors problem and reduce this system to the system y′ = By, φ′ = ω with constant coefficients.

This problem has already been examined by Mitropolśki and Samolenko but the calculations and details in their work are formidable and difficult to follow. Besides being simpler our method provides more precise estimates at all stages and can be extended to the differentiable case to provide a significant improvement over previous results.