Non-uniform beam-like structures have been widely used in engineering structures to achieve an optimal distribution of strength, stiffness and modes of vibration to satisfy design requirements. In this paper, a new asymptotic method for the analysis of these systems is presented. The linear partial differential equation which governs the response of the beam and the inhomogeneous boundary conditions have been put into a state space form. Choosing the small parameterεas a perturbation parameter, the asymptotic solutions are determined by the state space technique. Numerical examples are provided to illustrate the efficiency of the method.