The non-degeneracy of differential equations provides an effective tool for characterizing the existence and uniqueness of periodic solutions to nonlinear differential equations. However, research on the non-degeneracy of higher order differential equations remains relatively limited, and the problem of algorithmic and symbolic treatment of non-degeneracy has yet to be addressed. This paper investigates the non-degeneracy for a class of higher order ODEs in the space of bounded functions. First, we develop a connection between the non-degeneracy of such differential equations and the exponential dichotomy of their associated linear differential systems. This connection allows us to detect the non-degeneracy of a considered equation by analyzing the exponential dichotomy of its associated linear system. Then, we present an algebraic criterion to determine whether all the complex roots of a univariate polynomial have non-zero real parts using the properties of Hurwitz determinants and subresultant sequences. This criterion is used to derive critical algebraic conditions for analyzing the exponential dichotomy of a linear differential system. Next, using these algebraic conditions we reduce the problem of detecting the non-degeneracy of a wide class of n-th order ODEs to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by quantifier elimination. Finally, several examples are presented to demonstrate the effectiveness of the proposed approach.