  This work is devoted to giving a geometric framework for describing
higher-order non-autonomous mechanical systems. The starting point is to extend
the Lagrangian-Hamiltonian unified formalism of Skinner and Rusk for these
kinds of systems, generalizing previous developments for higher-order
autonomous mechanical systems and first-order non-autonomous mechanical
systems. Then, we use this unified formulation to derive the standard
Lagrangian and Hamiltonian formalisms, including the Legendre-Ostrogradsky map
and the Euler-Lagrange and the Hamilton equations, both for regular and
singular systems. As applications of our model, two examples of regular and
singular physical systems are studied.
