Motivated from the recent work of Srivastava et al. (H.\,M. Srivastava, O. Altıntas¸, S.\,K. Serenbay, Coefficient bounds for certain subclasses of starlike functions of complex order, Appl. Math. Lett. 24(2011)1359-1363.), we aim to determine the coefficient estimates for functions in certain subclasses of close-to-convex and related functions of complex order, which are here defined by means of Cauchy--Euler type non-homogeneous differential equation. Several interesting consequences of our results are also observed.