Bekiryazıcı, Zafer2022-12-262022-12-262022https://doi.org/10.31801/cfsuasmas.845845http://hdl.handle.net/20.500.12575/86476In this paper, we give a generalization of the osculating curves to the n -dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the ( n − 3 ) th binormal vector in the n-dimensional Euclidean space, which has been called ”a generalized osculating curve of type ( n − 3 ) ”. We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n . In particular, we characterize the osculating curves in E n in terms of their curvature functions. Finally, we show that the ratio of the ( n − 1 ) th and ( n − 2 ) th curvatures of the osculating curve is the solution of an ( n − 2 ) th order linear nonhomogeneous differential equation.enOsculating curve, curvatures, unit speed curve, higher order linear differential equationArticle7112122221303-5991