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This paper describes a new trajectory planning method for the output tracking control of linear flexible systems; this method computes the exact solution of the equilibrium manifold. We establish a system of differential equations by combining kinematic and dynamic constraints and reformulate them in a singularly perturbed dynamics to obtain the equilibrium manifold in the form of an infinite series, which is our planned trajectory. We show that for the desired output defined by exponential functions, the equilibrium manifold becomes a converging geometric series that has a succinct form of summation. In addition, the inverse torque, which is the feedforward command, is easily produced by using the computed exact equilibrium manifold. We validate the effectiveness of the proposed method through simulations and experimental studies using a single-link flexible arm.