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This paper proposes a method to determine trajectories of dynamic systems that steer between two end points while satisfying linear inequality constraints arising from limits on states and inputs. The method exploits the structure of the dynamic system written in a higher-order form to explicitly eliminate the state equations. The feasible trajectories of the dynamic system are sought within a characterization with a finite sum of mode functions. In this paper, the linear inequalities on inputs and states are replaced by a finite set of linear inequalities on the mode coefficients. This step changes the problem of trajectory generation into finding a convex polytope enclosed by the linear inequalities on the mode coefficients. A procedure is then developed to efficiently find the vertices of this bounding polytope. It is demonstrated in this paper that this method can generate feasible trajectories of the system in real-time and can quickly update the trajectories as the terminal conditions are changed. The procedure is demonstrated numerically by two examples. The results of one of the examples is implemented in hardware to explore the issues of real-time planning and control.