(untitled) (Communications in Mathematical Research, 2019)
In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ax^+−bx^−+ ϕ(x) = p(t), where a ≠ b are two positive constants and ϕ(s), p(t) are real analytic functions. Moreover, the p(t) is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions, the quasi-periodic oscillator has the Lagrange stability.