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In this paper, we will derive some twist criteria for the periodic solution of a periodic scalar Newtonian equation using the third order approximation. As an application to the forced pendulum x ̈ + ω^2 sin x = p(t), we will find an explicit bound P (ω) for the L1 norm, ∥p∥1, of the periodicforcingp(t)usingthefrequencyωasaparametersuchthattheleastamplitudeperiodic solution of the forced pendulum is of twist type when ∥p∥1 < P (ω). The bound P (ω) has the order of O(ω1/2) when ω is bounded away from resonance of orders ≤ 4 and ω → +∞.

Differential Galois theory has played important roles in the the- ory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order non- linear differential equation. We will define for the differential equation the differential Galois group, will study the structure of the group, and will prove the equivalent between the existence of the Liouvillian first integral and the solvability of the corresponding differential Galois group.