I also think of that as the correct answer, but i can't really extract the correct classical action of the system from the general solution of the classical path. I guess i do need more knowledge when it comes to integration following a long and tedious job.
In quantum mechanics, one of the major concerns is the propagator determination of the system. The propagator is completely expressed in terms of its classical in the Van-Vleck Pauli Formula.
In a harmonic oscillator in a time dependent force, the Lagrangian is given by...